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Find the effective annual yield of a bank account that pays interest at a rate of 7%, compounded daily; that is, divide the difference between the final and initial balances by the initial balance.
|
7.25
|
7.25
|
page 130-7
|
%
|
diff
|
math
|
||
Consider a tank used in certain hydrodynamic experiments. After one experiment the tank contains $200 \mathrm{~L}$ of a dye solution with a concentration of $1 \mathrm{~g} / \mathrm{L}$. To prepare for the next experiment, the tank is to be rinsed with fresh water flowing in at a rate of $2 \mathrm{~L} / \mathrm{min}$, the well-stirred solution flowing out at the same rate. Find the time that will elapse before the concentration of dye in the tank reaches $1 \%$ of its original value.
|
460.5
|
460.5
|
Page 59-1
|
min
|
diff
|
math
|
||
A certain vibrating system satisfies the equation $u^{\prime \prime}+\gamma u^{\prime}+u=0$. Find the value of the damping coefficient $\gamma$ for which the quasi period of the damped motion is $50 \%$ greater than the period of the corresponding undamped motion.
|
1.4907
|
1.4907
|
page203-13
|
diff
|
math
|
|||
Find the value of $y_0$ for which the solution of the initial value problem
$$
y^{\prime}-y=1+3 \sin t, \quad y(0)=y_0
$$
remains finite as $t \rightarrow \infty$
|
-2.5
|
-2.5
|
Page 40-30
|
diff
|
math
|
|||
A certain spring-mass system satisfies the initial value problem
$$
u^{\prime \prime}+\frac{1}{4} u^{\prime}+u=k g(t), \quad u(0)=0, \quad u^{\prime}(0)=0
$$
where $g(t)=u_{3 / 2}(t)-u_{5 / 2}(t)$ and $k>0$ is a parameter.
Suppose $k=2$. Find the time $\tau$ after which $|u(t)|<0.1$ for all $t>\tau$.
|
25.6773
|
25.6773
|
page336-16
|
diff
|
math
|
|||
Suppose that a sum $S_0$ is invested at an annual rate of return $r$ compounded continuously.
Determine $T$ if $r=7 \%$.
|
9.90
|
9.90
|
page 60-7
|
year
|
diff
|
math
|
||
A mass weighing $2 \mathrm{lb}$ stretches a spring 6 in. If the mass is pulled down an additional 3 in. and then released, and if there is no damping, by determining the position $u$ of the mass at any time $t$, find the frequency of the motion
|
$\pi/4$
|
0.7854
|
page202-5
|
s
|
diff
|
math
|
||
If $\mathbf{x}=\left(\begin{array}{c}2 \\ 3 i \\ 1-i\end{array}\right)$ and $\mathbf{y}=\left(\begin{array}{c}-1+i \\ 2 \\ 3-i\end{array}\right)$, find $(\mathbf{y}, \mathbf{y})$.
|
16
|
16
|
page372-8
|
diff
|
math
|
|||
Consider the initial value problem
$$
4 y^{\prime \prime}+12 y^{\prime}+9 y=0, \quad y(0)=1, \quad y^{\prime}(0)=-4 .
$$
Determine where the solution has the value zero.
|
0.4
|
0.4
|
page172-15
|
diff
|
math
|
|||
A certain college graduate borrows $8000 to buy a car. The lender charges interest at an annual rate of 10%. What monthly payment rate is required to pay off the loan in 3 years?
|
258.14
|
258.14
|
page131-9
|
$
|
diff
|
math
|
||
Consider the initial value problem
$$
y^{\prime \prime}+\gamma y^{\prime}+y=k \delta(t-1), \quad y(0)=0, \quad y^{\prime}(0)=0
$$
where $k$ is the magnitude of an impulse at $t=1$ and $\gamma$ is the damping coefficient (or resistance).
Let $\gamma=\frac{1}{2}$. Find the value of $k$ for which the response has a peak value of 2 ; call this value $k_1$.
|
2.8108
|
2.8108
|
page344-15
|
diff
|
math
|
|||
If a series circuit has a capacitor of $C=0.8 \times 10^{-6} \mathrm{~F}$ and an inductor of $L=0.2 \mathrm{H}$, find the resistance $R$ so that the circuit is critically damped.
|
1000
|
1000
|
page203-18
|
$\Omega$
|
diff
|
math
|
||
If $y_1$ and $y_2$ are a fundamental set of solutions of $t y^{\prime \prime}+2 y^{\prime}+t e^t y=0$ and if $W\left(y_1, y_2\right)(1)=2$, find the value of $W\left(y_1, y_2\right)(5)$.
|
2/25
|
0.08
|
page156-34
|
diff
|
math
|
|||
Consider the initial value problem
$$
5 u^{\prime \prime}+2 u^{\prime}+7 u=0, \quad u(0)=2, \quad u^{\prime}(0)=1
$$
Find the smallest $T$ such that $|u(t)| \leq 0.1$ for all $t>T$.
|
14.5115
|
14.5115
|
page163-24
|
diff
|
math
|
|||
Consider the initial value problem
$$
y^{\prime}=t y(4-y) / 3, \quad y(0)=y_0
$$
Suppose that $y_0=0.5$. Find the time $T$ at which the solution first reaches the value 3.98.
|
3.29527
|
3.29527
|
Page 49 27
|
diff
|
math
|
|||
Consider the initial value problem
$$
2 y^{\prime \prime}+3 y^{\prime}-2 y=0, \quad y(0)=1, \quad y^{\prime}(0)=-\beta,
$$
where $\beta>0$.
Find the smallest value of $\beta$ for which the solution has no minimum point.
|
2
|
2
|
page144-25
|
diff
|
math
|
|||
Consider the initial value problem
$$
y^{\prime}=t y(4-y) /(1+t), \quad y(0)=y_0>0 .
$$
If $y_0=2$, find the time $T$ at which the solution first reaches the value 3.99.
|
2.84367
|
2.84367
|
Page 49 28
|
diff
|
math
|
|||
A mass of $0.25 \mathrm{~kg}$ is dropped from rest in a medium offering a resistance of $0.2|v|$, where $v$ is measured in $\mathrm{m} / \mathrm{s}$.
If the mass is to attain a velocity of no more than $10 \mathrm{~m} / \mathrm{s}$, find the maximum height from which it can be dropped.
|
13.45
|
13.45
|
page66-28
|
m
|
diff
|
math
|
||
A home buyer can afford to spend no more than $\$ 800$ /month on mortgage payments. Suppose that the interest rate is $9 \%$ and that the term of the mortgage is 20 years. Assume that interest is compounded continuously and that payments are also made continuously.
Determine the maximum amount that this buyer can afford to borrow.
|
89,034.79
|
89,034.79
|
page 61-10
|
$
|
diff
|
math
|
||
A spring is stretched 6 in by a mass that weighs $8 \mathrm{lb}$. The mass is attached to a dashpot mechanism that has a damping constant of $0.25 \mathrm{lb} \cdot \mathrm{s} / \mathrm{ft}$ and is acted on by an external force of $4 \cos 2 t \mathrm{lb}$.
If the given mass is replaced by a mass $m$, determine the value of $m$ for which the amplitude of the steady state response is maximum.
|
4
|
4
|
page216-11
|
slugs
|
diff
|
math
|
||
A recent college graduate borrows $\$ 100,000$ at an interest rate of $9 \%$ to purchase a condominium. Anticipating steady salary increases, the buyer expects to make payments at a monthly rate of $800(1+t / 120)$, where $t$ is the number of months since the loan was made.
Assuming that this payment schedule can be maintained, when will the loan be fully paid?
|
135.36
|
135.36
|
page61-11
|
months
|
diff
|
math
|
||
Consider the initial value problem
$$
y^{\prime}+\frac{1}{4} y=3+2 \cos 2 t, \quad y(0)=0
$$
Determine the value of $t$ for which the solution first intersects the line $y=12$.
|
10.065778
|
10.065778
|
Page 40 29
|
diff
|
math
|
|||
An investor deposits $1000 in an account paying interest at a rate of 8% compounded monthly, and also makes additional deposits of \$25 per month. Find the balance in the account after 3 years.
|
2283.63
|
2283.63
|
page 131-8
|
$
|
diff
|
math
|
||
A mass of $0.25 \mathrm{~kg}$ is dropped from rest in a medium offering a resistance of $0.2|v|$, where $v$ is measured in $\mathrm{m} / \mathrm{s}$.
If the mass is dropped from a height of $30 \mathrm{~m}$, find its velocity when it hits the ground.
|
11.58
|
11.58
|
page 66-28
|
m/s
|
diff
|
math
|
||
A mass of $100 \mathrm{~g}$ stretches a spring $5 \mathrm{~cm}$. If the mass is set in motion from its equilibrium position with a downward velocity of $10 \mathrm{~cm} / \mathrm{s}$, and if there is no damping, determine when does the mass first return to its equilibrium position.
|
$\pi/14$
|
0.2244
|
page202-6
|
s
|
diff
|
math
|
||
Suppose that a tank containing a certain liquid has an outlet near the bottom. Let $h(t)$ be the height of the liquid surface above the outlet at time $t$. Torricelli's principle states that the outflow velocity $v$ at the outlet is equal to the velocity of a particle falling freely (with no drag) from the height $h$.
Consider a water tank in the form of a right circular cylinder that is $3 \mathrm{~m}$ high above the outlet. The radius of the tank is $1 \mathrm{~m}$ and the radius of the circular outlet is $0.1 \mathrm{~m}$. If the tank is initially full of water, determine how long it takes to drain the tank down to the level of the outlet.
|
130.41
|
130.41
|
page 60-6
|
s
|
diff
|
math
|
||
Solve the initial value problem $y^{\prime \prime}-y^{\prime}-2 y=0, y(0)=\alpha, y^{\prime}(0)=2$. Then find $\alpha$ so that the solution approaches zero as $t \rightarrow \infty$.
|
−2
|
−2
|
page144-21
|
diff
|
math
|
|||
If $y_1$ and $y_2$ are a fundamental set of solutions of $t^2 y^{\prime \prime}-2 y^{\prime}+(3+t) y=0$ and if $W\left(y_1, y_2\right)(2)=3$, find the value of $W\left(y_1, y_2\right)(4)$.
|
4.946
|
4.946
|
page156-35
|
diff
|
math
|
|||
Radium-226 has a half-life of 1620 years. Find the time period during which a given amount of this material is reduced by one-quarter.
|
672.4
|
672.4
|
Page 17 14
|
Year
|
diff
|
math
|
||
A tank originally contains $100 \mathrm{gal}$ of fresh water. Then water containing $\frac{1}{2} \mathrm{lb}$ of salt per gallon is poured into the tank at a rate of $2 \mathrm{gal} / \mathrm{min}$, and the mixture is allowed to leave at the same rate. After $10 \mathrm{~min}$ the process is stopped, and fresh water is poured into the tank at a rate of $2 \mathrm{gal} / \mathrm{min}$, with the mixture again leaving at the same rate. Find the amount of salt in the tank at the end of an additional $10 \mathrm{~min}$.
|
7.42
|
7.42
|
Page 60-3
|
lb
|
diff
|
math
|
||
A young person with no initial capital invests $k$ dollars per year at an annual rate of return $r$. Assume that investments are made continuously and that the return is compounded continuously.
If $r=7.5 \%$, determine $k$ so that $\$ 1$ million will be available for retirement in 40 years.
|
3930
|
3930
|
page 60-8
|
$
|
diff
|
math
|
||
Consider the initial value problem
$$
y^{\prime \prime}+2 a y^{\prime}+\left(a^2+1\right) y=0, \quad y(0)=1, \quad y^{\prime}(0)=0 .
$$
For $a=1$ find the smallest $T$ such that $|y(t)|<0.1$ for $t>T$.
|
1.8763
|
1.8763
|
page164-26
|
diff
|
math
|
|||
Consider the initial value problem
$$
y^{\prime \prime}+\gamma y^{\prime}+y=\delta(t-1), \quad y(0)=0, \quad y^{\prime}(0)=0,
$$
where $\gamma$ is the damping coefficient (or resistance).
Find the time $t_1$ at which the solution attains its maximum value.
|
2.3613
|
2.3613
|
page344-14
|
diff
|
math
|
|||
Consider the initial value problem
$$
y^{\prime}+\frac{2}{3} y=1-\frac{1}{2} t, \quad y(0)=y_0 .
$$
Find the value of $y_0$ for which the solution touches, but does not cross, the $t$-axis.
|
−1.642876
|
−1.642876
|
Page 40 28
|
diff
|
math
|
|||
A radioactive material, such as the isotope thorium-234, disintegrates at a rate proportional to the amount currently present. If $Q(t)$ is the amount present at time $t$, then $d Q / d t=-r Q$, where $r>0$ is the decay rate. If $100 \mathrm{mg}$ of thorium-234 decays to $82.04 \mathrm{mg}$ in 1 week, determine the decay rate $r$.
|
0.02828
|
0.02828
|
Section 1.2, page 15 12. (a)
|
$\text{day}^{-1}$
|
diff
|
math
|
||
Newton's law of cooling states that the temperature of an object changes at a rate proportional to the difference between its temperature and that of its surroundings. Suppose that the temperature of a cup of coffee obeys Newton's law of cooling. If the coffee has a temperature of $200^{\circ} \mathrm{F}$ when freshly poured, and $1 \mathrm{~min}$ later has cooled to $190^{\circ} \mathrm{F}$ in a room at $70^{\circ} \mathrm{F}$, determine when the coffee reaches a temperature of $150^{\circ} \mathrm{F}$.
|
6.07
|
6.07
|
page62-16
|
min
|
diff
|
math
|
||
Solve the initial value problem $4 y^{\prime \prime}-y=0, y(0)=2, y^{\prime}(0)=\beta$. Then find $\beta$ so that the solution approaches zero as $t \rightarrow \infty$.
|
-1
|
-1
|
page144-22
|
diff
|
math
|
|||
Consider the initial value problem (see Example 5)
$$
y^{\prime \prime}+5 y^{\prime}+6 y=0, \quad y(0)=2, \quad y^{\prime}(0)=\beta
$$
where $\beta>0$.
Determine the smallest value of $\beta$ for which $y_m \geq 4$.
|
16.3923
|
16.3923
|
page145-26
|
diff
|
math
|
|||
A home buyer can afford to spend no more than $\$ 800 /$ month on mortgage payments. Suppose that the interest rate is $9 \%$ and that the term of the mortgage is 20 years. Assume that interest is compounded continuously and that payments are also made continuously.
Determine the total interest paid during the term of the mortgage.
|
102,965.21
|
102,965.21
|
page61-10
|
$
|
diff
|
math
|
||
Find the fundamental period of the given function:
$$f(x)=\left\{\begin{array}{ll}(-1)^n, & 2 n-1 \leq x<2 n, \\ 1, & 2 n \leq x<2 n+1 ;\end{array} \quad n=0, \pm 1, \pm 2, \ldots\right.$$
|
4
|
4
|
page593-8
|
diff
|
math
|
|||
A homebuyer wishes to finance the purchase with a \$95,000 mortgage with a 20-year term. What is the maximum interest rate the buyer can afford if the monthly payment is not to exceed \$900?
|
9.73
|
9.73
|
page131-13
|
%
|
diff
|
math
|
||
A homebuyer wishes to take out a mortgage of $100,000 for a 30-year period. What monthly payment is required if the interest rate is 9%?
|
804.62
|
804.62
|
page131-10
|
$
|
diff
|
math
|
||
Let a metallic rod $20 \mathrm{~cm}$ long be heated to a uniform temperature of $100^{\circ} \mathrm{C}$. Suppose that at $t=0$ the ends of the bar are plunged into an ice bath at $0^{\circ} \mathrm{C}$, and thereafter maintained at this temperature, but that no heat is allowed to escape through the lateral surface. Determine the temperature at the center of the bar at time $t=30 \mathrm{~s}$ if the bar is made of silver.
|
35.91
|
35.91
|
page619-18
|
${ }^{\circ} \mathrm{C}$
|
diff
|
math
|
||
Find $\gamma$ so that the solution of the initial value problem $x^2 y^{\prime \prime}-2 y=0, y(1)=1, y^{\prime}(1)=\gamma$ is bounded as $x \rightarrow 0$.
|
2
|
2
|
page277-37
|
diff
|
math
|
|||
A tank contains 100 gal of water and $50 \mathrm{oz}$ of salt. Water containing a salt concentration of $\frac{1}{4}\left(1+\frac{1}{2} \sin t\right) \mathrm{oz} / \mathrm{gal}$ flows into the tank at a rate of $2 \mathrm{gal} / \mathrm{min}$, and the mixture in the tank flows out at the same rate.
The long-time behavior of the solution is an oscillation about a certain constant level. What is the amplitude of the oscillation?
|
0.24995
|
0.24995
|
Page 60-5
|
diff
|
math
|
|||
A mass weighing $8 \mathrm{lb}$ stretches a spring 1.5 in. The mass is also attached to a damper with coefficient $\gamma$. Determine the value of $\gamma$ for which the system is critically damped; be sure to give the units for $\gamma$
|
8
|
8
|
page203-17
|
$\mathrm{lb} \cdot \mathrm{s} / \mathrm{ft}$
|
diff
|
math
|
||
Your swimming pool containing 60,000 gal of water has been contaminated by $5 \mathrm{~kg}$ of a nontoxic dye that leaves a swimmer's skin an unattractive green. The pool's filtering system can take water from the pool, remove the dye, and return the water to the pool at a flow rate of $200 \mathrm{gal} / \mathrm{min}$. Find the time $T$ at which the concentration of dye first reaches the value $0.02 \mathrm{~g} / \mathrm{gal}$.
|
7.136
|
7.136
|
Page 18 19
|
hour
|
diff
|
math
|
||
For small, slowly falling objects, the assumption made in the text that the drag force is proportional to the velocity is a good one. For larger, more rapidly falling objects, it is more accurate to assume that the drag force is proportional to the square of the velocity. If m = 10 kg, find the drag coefficient so that the limiting velocity is 49 m/s.
|
$\frac{2}{49}$
|
0.0408
|
1 25(c)
|
diff
|
math
|
|||
Consider the initial value problem
$$
3 u^{\prime \prime}-u^{\prime}+2 u=0, \quad u(0)=2, \quad u^{\prime}(0)=0
$$
For $t>0$ find the first time at which $|u(t)|=10$.
|
10.7598
|
10.7598
|
page163-23
|
diff
|
math
|
|||
Consider the initial value problem
$$
9 y^{\prime \prime}+12 y^{\prime}+4 y=0, \quad y(0)=a>0, \quad y^{\prime}(0)=-1
$$
Find the critical value of $a$ that separates solutions that become negative from those that are always positive.
|
1.5
|
1.5
|
page172-18
|
diff
|
math
|
|||
We have $S_0=0, r=0.08$, and $k=\$ 2000$, and we wish to determine $S(40)$. From Eq. $$
S(t)=S_0 e^{r t}+(k / r)\left(e^{r t}-1\right)
$$ we have
$$
S(40)=(25,000)\left(e^{3.2}-1\right)=\$ 588,313
$$
|
For instance, suppose that one opens an individual retirement account (IRA) at age 25 and makes annual investments of $\$ 2000$ thereafter in a continuous manner. Assuming a rate of return of $8 \%$, what will be the balance in the IRA at age 65 ?
|
588313
|
588313
|
2.3.2
|
$\$$
|
diff
|
math
|
|
The spring constant is $k=10 \mathrm{lb} / 2 \mathrm{in} .=60 \mathrm{lb} / \mathrm{ft}$, and the mass is $m=w / g=10 / 32 \mathrm{lb} \cdot \mathrm{s}^2 / \mathrm{ft}$. Hence the equation of motion reduces to
$$
u^{\prime \prime}+192 u=0
$$
and the general solution is
$$
u=A \cos (8 \sqrt{3} t)+B \sin (8 \sqrt{3} t)
$$
The solution satisfying the initial conditions $u(0)=1 / 6 \mathrm{ft}$ and $u^{\prime}(0)=-1 \mathrm{ft} / \mathrm{s}$ is
$$
u=\frac{1}{6} \cos (8 \sqrt{3} t)-\frac{1}{8 \sqrt{3}} \sin (8 \sqrt{3} t)
$$
The natural frequency is $\omega_0=\sqrt{192} \cong 13.856 \mathrm{rad} / \mathrm{s}$, so the period is $T=2 \pi / \omega_0 \cong 0.45345 \mathrm{~s}$. The amplitude $R$ and phase $\delta$ are found from Eqs. $$
R=\sqrt{A^2+B^2}, \quad \tan \delta=B / A
$$. We have
$$
R^2=\frac{1}{36}+\frac{1}{192}=\frac{19}{576}, \quad \text { so } \quad R \cong 0.18162 \mathrm{ft}
$$
|
Suppose that a mass weighing $10 \mathrm{lb}$ stretches a spring $2 \mathrm{in}$. If the mass is displaced an additional 2 in. and is then set in motion with an initial upward velocity of $1 \mathrm{ft} / \mathrm{s}$, by determining the position of the mass at any later time, calculate the amplitude of the motion.
|
0.18162
|
0.18162
|
3.7.2
|
$\mathrm{ft}$
|
diff
|
math
|
|
We assume that salt is neither created nor destroyed in the tank. Therefore variations in the amount of salt are due solely to the flows in and out of the tank. More precisely, the rate of change of salt in the tank, $d Q / d t$, is equal to the rate at which salt is flowing in minus the rate at which it is flowing out. In symbols,
$$
\frac{d Q}{d t}=\text { rate in }- \text { rate out }
$$
The rate at which salt enters the tank is the concentration $\frac{1}{4} \mathrm{lb} / \mathrm{gal}$ times the flow rate $r \mathrm{gal} / \mathrm{min}$, or $(r / 4) \mathrm{lb} / \mathrm{min}$. To find the rate at which salt leaves the tankl we need to multiply the concentration of salt in the tank by the rate of outflow, $r \mathrm{gal} / \mathrm{min}$. Since the rates of flow in and out are equal, the volume of water in the tank remains constant at $100 \mathrm{gal}$, and since the mixture is "well-stirred," the concentration throughout the tank is the same, namely, $[Q(t) / 100] \mathrm{lb} / \mathrm{gal}$. Therefore the rate at which salt leaves the tank is $[r Q(t) / 100] \mathrm{lb} / \mathrm{min}$. Thus the differential equation governing this process is
$$
\frac{d Q}{d t}=\frac{r}{4}-\frac{r Q}{100}
$$
The initial condition is
$$
Q(0)=Q_0
$$
Upon thinking about the problem physically, we might anticipate that eventually the mixture originally in the tank will be essentially replaced by the mixture flowing in, whose concentration is $\frac{1}{4} \mathrm{lb} / \mathrm{gal}$. Consequently, we might expect that ultimately the amount of salt in the tank would be very close to $25 \mathrm{lb}$. We can also find the limiting amount $Q_L=25$ by setting $d Q / d t$ equal to zero in the equation and solving the resulting algebraic equation for $Q$. Rewriting the above equation in the standard form for a linear equation, we have
$$
\frac{d Q}{d t}+\frac{r Q}{100}=\frac{r}{4}
$$
Thus the integrating factor is $e^{r t / 100}$ and the general solution is
$$
Q(t)=25+c e^{-r t / 100}
$$
where $c$ is an arbitrary constant. To satisfy the initial condition, we must choose $c=Q_0-25$. Therefore the solution of the initial value problem is
$$
Q(t)=25+\left(Q_0-25\right) e^{-r t / 100}
$$
or
$$
Q(t)=25\left(1-e^{-r t / 100}\right)+Q_0 e^{-r t / 100}
$$
From Eq., you can see that $Q(t) \rightarrow 25$ (lb) as $t \rightarrow \infty$, so the limiting value $Q_L$ is 25 , confirming our physical intuition. Further, $Q(t)$ approaches the limit more rapidly as $r$ increases. In interpreting the solution, note that the second term on the right side is the portion of the original salt that remains at time $t$, while the first term gives the amount of salt in the tank due to the action of the flow processes. Now suppose that $r=3$ and $Q_0=2 Q_L=50$; then Eq. becomes
$$
Q(t)=25+25 e^{-0.03 t}
$$
Since $2 \%$ of 25 is 0.5 , we wish to find the time $T$ at which $Q(t)$ has the value 25.5. Substituting $t=T$ and $Q=25.5$ in Eq. (8) and solving for $T$, we obtain
$$
T=(\ln 50) / 0.03 \cong 130.400766848(\mathrm{~min}) .
$$
|
At time $t=0$ a tank contains $Q_0 \mathrm{lb}$ of salt dissolved in 100 gal of water. Assume that water containing $\frac{1}{4} \mathrm{lb}$ of salt/gal is entering the tank at a rate of $r \mathrm{gal} / \mathrm{min}$ and that the well-stirred mixture is draining from the tank at the same rate. Set up the initial value problem that describes this flow process. By finding the amount of salt $Q(t)$ in the tank at any time, and the limiting amount $Q_L$ that is present after a very long time, if $r=3$ and $Q_0=2 Q_L$, find the time $T$ after which the salt level is within $2 \%$ of $Q_L$.
|
$(\ln 50) / 0.03$
|
130.400766848
|
2.3.1
|
$\mathrm{~min}$
|
diff
|
math
|
|
The spring constant is $k=10 \mathrm{lb} / 2 \mathrm{in} .=60 \mathrm{lb} / \mathrm{ft}$, and the mass is $m=w / g=10 / 32 \mathrm{lb} \cdot \mathrm{s}^2 / \mathrm{ft}$. Hence the equation of motion reduces to
$$
u^{\prime \prime}+192 u=0
$$
and the general solution is
$$
u=A \cos (8 \sqrt{3} t)+B \sin (8 \sqrt{3} t)
$$
The solution satisfying the initial conditions $u(0)=1 / 6 \mathrm{ft}$ and $u^{\prime}(0)=-1 \mathrm{ft} / \mathrm{s}$ is
$$
u=\frac{1}{6} \cos (8 \sqrt{3} t)-\frac{1}{8 \sqrt{3}} \sin (8 \sqrt{3} t)
$$
The natural frequency is $\omega_0=\sqrt{192} \cong 13.856 \mathrm{rad} / \mathrm{s}$, so the period is $T=2 \pi / \omega_0 \cong 0.45345 \mathrm{~s}$. The amplitude $R$ and phase $\delta$ are found from Eqs. (17) $$R=\sqrt{A^2+B^2}, \quad \tan \delta=B / A$$. We have
$$
R^2=\frac{1}{36}+\frac{1}{192}=\frac{19}{576}, \quad \text { so } \quad R \cong 0.18162 \mathrm{ft}
$$
The second of Eqs. (17) yields $\tan \delta=-\sqrt{3} / 4$. There are two solutions of this equation, one in the second quadrant and one in the fourth. In the present problem $\cos \delta>0$ and $\sin \delta<0$, so $\delta$ is in the fourth quadrant, namely,
$$
\delta=-\arctan (\sqrt{3} / 4) \cong-0.40864 \mathrm{rad}
$$
|
Suppose that a mass weighing $10 \mathrm{lb}$ stretches a spring $2 \mathrm{in}$. If the mass is displaced an additional 2 in. and is then set in motion with an initial upward velocity of $1 \mathrm{ft} / \mathrm{s}$, by determining the position of the mass at any later time, calculate the phase of the motion.
|
$-\arctan (\sqrt{3} / 4)$
|
-0.40864
|
3.7.2
|
$\mathrm{rad}$
|
diff
|
math
|
|
It is convenient to scale the solution (11) $$y=\frac{y_0 K}{y_0+\left(K-y_0\right) e^{-r t}}
$$ to the carrying capacity $K$; thus we write Eq. (11) in the form
$$
\frac{y}{K}=\frac{y_0 / K}{\left(y_0 / K\right)+\left[1-\left(y_0 / K\right)\right] e^{-r t}}
$$
Using the data given in the problem, we find that
$$
\frac{y(2)}{K}=\frac{0.25}{0.25+0.75 e^{-1.42}} \cong 0.5797 .
$$
Consequently, $y(2) \cong 46.7 \times 10^6 \mathrm{~kg}$.
|
The logistic model has been applied to the natural growth of the halibut population in certain areas of the Pacific Ocean. ${ }^{12}$ Let $y$, measured in kilograms, be the total mass, or biomass, of the halibut population at time $t$. The parameters in the logistic equation are estimated to have the values $r=0.71 /$ year and $K=80.5 \times 10^6 \mathrm{~kg}$. If the initial biomass is $y_0=0.25 K$, find the biomass 2 years later.
|
46.7
|
46.7
|
2.5.1
|
$10^6 \mathrm{~kg}$
|
diff
|
math
|
|
If the distribution of $Y$ is $b(n, 0.25)$, give a lower bound for $P(|Y / n-0.25|<0.05)$ when $n=100$.
|
$0.25$
|
0.25
|
5.8-5 (a)
|
stat
|
math
|
|||
A device contains three components, each of which has a lifetime in hours with the pdf
$$
f(x)=\frac{2 x}{10^2} e^{-(x / 10)^2}, \quad 0 < x < \infty .
$$
The device fails with the failure of one of the components. Assuming independent lifetimes, what is the probability that the device fails in the first hour of its operation? HINT: $G(y)=P(Y \leq y)=1-P(Y>y)=1-P$ (all three $>y$ ).
|
$0.03$
|
0.03
|
5.3-13
|
stat
|
math
|
|||
The tensile strength $X$ of paper, in pounds per square inch, has $\mu=30$ and $\sigma=3$. A random sample of size $n=100$ is taken from the distribution of tensile strengths. Compute the probability that the sample mean $\bar{X}$ is greater than 29.5 pounds per square inch.
|
$0.9522$
|
0.9522
|
5.6-13
|
stat
|
math
|
|||
Let $\bar{X}$ be the mean of a random sample of size 36 from an exponential distribution with mean 3 . Approximate $P(2.5 \leq \bar{X} \leq 4)$
|
$0.8185$
|
0.8185
|
5.6-3
|
stat
|
math
|
|||
Let $X_1, X_2$ be a random sample of size $n=2$ from a distribution with pdf $f(x)=3 x^2, 0 < x < 1$. Determine $P\left(\max X_i < 3 / 4\right)=P\left(X_1<3 / 4, X_2<3 / 4\right)$
|
$\frac{729}{4096}$
|
0.178
|
5.3-9
|
stat
|
math
|
|||
Let $X$ equal the tarsus length for a male grackle. Assume that the distribution of $X$ is $N(\mu, 4.84)$. Find the sample size $n$ that is needed so that we are $95 \%$ confident that the maximum error of the estimate of $\mu$ is 0.4 .
|
$117$
|
117
|
7.4-1
|
stat
|
math
|
|||
In a study concerning a new treatment of a certain disease, two groups of 25 participants in each were followed for five years. Those in one group took the old treatment and those in the other took the new treatment. The theoretical dropout rate for an individual was $50 \%$ in both groups over that 5 -year period. Let $X$ be the number that dropped out in the first group and $Y$ the number in the second group. Assuming independence where needed, give the sum that equals the probability that $Y \geq X+2$. HINT: What is the distribution of $Y-X+25$ ?
|
$0.3359$
|
0.3359
|
5.4-17
|
stat
|
math
|
|||
Let $X$ and $Y$ have a bivariate normal distribution with correlation coefficient $\rho$. To test $H_0: \rho=0$ against $H_1: \rho \neq 0$, a random sample of $n$ pairs of observations is selected. Suppose that the sample correlation coefficient is $r=0.68$. Using a significance level of $\alpha=0.05$, find the smallest value of the sample size $n$ so that $H_0$ is rejected.
|
$9$
|
9
|
9.6-11
|
stat
|
math
|
|||
In order to estimate the proportion, $p$, of a large class of college freshmen that had high school GPAs from 3.2 to 3.6 , inclusive, a sample of $n=50$ students was taken. It was found that $y=9$ students fell into this interval. Give a point estimate of $p$.
|
$0.1800$
|
0.1800
|
7.3-5
|
stat
|
math
|
|||
If $\bar{X}$ and $\bar{Y}$ are the respective means of two independent random samples of the same size $n$, find $n$ if we want $\bar{x}-\bar{y} \pm 4$ to be a $90 \%$ confidence interval for $\mu_X-\mu_Y$. Assume that the standard deviations are known to be $\sigma_X=15$ and $\sigma_Y=25$.
|
$144$
|
144
|
7.4-15
|
stat
|
math
|
|||
For a public opinion poll for a close presidential election, let $p$ denote the proportion of voters who favor candidate $A$. How large a sample should be taken if we want the maximum error of the estimate of $p$ to be equal to 0.03 with $95 \%$ confidence?
|
$1068$
|
1068
|
7.4-7
|
stat
|
math
|
|||
Let the distribution of $T$ be $t(17)$. Find $t_{0.01}(17)$.
|
$2.567$
|
2.567
|
5.5-15 (a)
|
stat
|
math
|
|||
Let $X_1, X_2, \ldots, X_{16}$ be a random sample from a normal distribution $N(77,25)$. Compute $P(77<\bar{X}<79.5)$.
|
$0.4772$
|
0.4772
|
5.5-1
|
stat
|
math
|
|||
5.4-19. A doorman at a hotel is trying to get three taxicabs for three different couples. The arrival of empty cabs has an exponential distribution with mean 2 minutes. Assuming independence, what is the probability that the doorman will get all three couples taken care of within 6 minutes?
|
$0.5768$
|
0.5768
|
5.4-19
|
stat
|
math
|
|||
Consider the following two groups of women: Group 1 consists of women who spend less than $\$ 500$ annually on clothes; Group 2 comprises women who spend over $\$ 1000$ annually on clothes. Let $p_1$ and $p_2$ equal the proportions of women in these two groups, respectively, who believe that clothes are too expensive. If 1009 out of a random sample of 1230 women from group 1 and 207 out of a random sample 340 from group 2 believe that clothes are too expensive, Give a point estimate of $p_1-p_2$.
|
$0.2115$
|
0.2115
|
7.3-9
|
stat
|
math
|
|||
Given below example: Approximate $P(39.75 \leq \bar{X} \leq 41.25)$, where $\bar{X}$ is the mean of a random sample of size 32 from a distribution with mean $\mu=40$ and variance $\sigma^2=8$. In the above example, compute $P(1.7 \leq Y \leq 3.2)$ with $n=4$
|
$0.6749$
|
0.6749
|
5.6-9
|
stat
|
math
|
|||
If the distribution of $Y$ is $b(n, 0.25)$, give a lower bound for $P(|Y / n-0.25|<0.05)$ when $n=1000$.
|
$0.925$
|
0.925
|
5.8-5
|
stat
|
math
|
|||
Let $Y_1 < Y_2 < Y_3 < Y_4 < Y_5 < Y_6$ be the order statistics of a random sample of size $n=6$ from a distribution of the continuous type having $(100 p)$ th percentile $\pi_p$. Compute $P\left(Y_2 < \pi_{0.5} < Y_5\right)$.
|
$0.7812$
|
0.7812
|
7.5-1
|
stat
|
math
|
|||
Let $X_1, X_2$ be independent random variables representing lifetimes (in hours) of two key components of a
device that fails when and only when both components fail. Say each $X_i$ has an exponential distribution with mean 1000. Let $Y_1=\min \left(X_1, X_2\right)$ and $Y_2=\max \left(X_1, X_2\right)$, so that the space of $Y_1, Y_2$ is $ 0< y_1 < y_2 < \infty $ Find $G\left(y_1, y_2\right)=P\left(Y_1 \leq y_1, Y_2 \leq y_2\right)$.
|
0.5117
|
0.5117
|
5.2-13
|
stat
|
math
|
|||
Let $Z_1, Z_2, \ldots, Z_7$ be a random sample from the standard normal distribution $N(0,1)$. Let $W=Z_1^2+Z_2^2+$ $\cdots+Z_7^2$. Find $P(1.69 < W < 14.07)$
|
$0.925$
|
0.925
|
5.4-5
|
stat
|
math
|
|||
Let $X_1$ and $X_2$ be independent Poisson random variables with respective means $\lambda_1=2$ and $\lambda_2=3$. Find $P\left(X_1=3, X_2=5\right)$. HINT. Note that this event can occur if and only if $\left\{X_1=1, X_2=0\right\}$ or $\left\{X_1=0, X_2=1\right\}$.
|
0.0182
|
0.0182
|
5.3-1
|
stat
|
math
|
|||
Let $Y$ be the number of defectives in a box of 50 articles taken from the output of a machine. Each article is defective with probability 0.01 . Find the probability that $Y=0,1,2$, or 3 By using the binomial distribution.
|
$0.9984$
|
0.9984
|
5.9-1 (a)
|
stat
|
math
|
|||
Some dentists were interested in studying the fusion of embryonic rat palates by a standard transplantation technique. When no treatment is used, the probability of fusion equals approximately 0.89 . The dentists would like to estimate $p$, the probability of fusion, when vitamin A is lacking. How large a sample $n$ of rat embryos is needed for $y / n \pm 0.10$ to be a $95 \%$ confidence interval for $p$ ?
|
$38$
|
38
|
7.4-11
|
stat
|
math
|
|||
To determine the effect of $100 \%$ nitrate on the growth of pea plants, several specimens were planted and then watered with $100 \%$ nitrate every day. At the end of
two weeks, the plants were measured. Here are data on seven of them:
$$
\begin{array}{lllllll}
17.5 & 14.5 & 15.2 & 14.0 & 17.3 & 18.0 & 13.8
\end{array}
$$
Assume that these data are a random sample from a normal distribution $N\left(\mu, \sigma^2\right)$. Find the value of a point estimate of $\mu$.
|
$15.757$
|
15.757
|
7.1-3
|
stat
|
math
|
|||
Suppose that the distribution of the weight of a prepackaged '1-pound bag' of carrots is $N\left(1.18,0.07^2\right)$ and the distribution of the weight of a prepackaged '3-pound bag' of carrots is $N\left(3.22,0.09^2\right)$. Selecting bags at random, find the probability that the sum of three 1-pound bags exceeds the weight of one 3-pound bag. HInT: First determine the distribution of $Y$, the sum of the three, and then compute $P(Y>W)$, where $W$ is the weight of the 3-pound bag.
|
$0.9830$
|
0.9830
|
5.5-7
|
stat
|
math
|
|||
The distributions of incomes in two cities follow the two Pareto-type pdfs $$ f(x)=\frac{2}{x^3}, 1 < x < \infty , \text { and } g(y)= \frac{3}{y^4} , \quad 1 < y < \infty,$$ respectively. Here one unit represents $ 20,000$. One person with income is selected at random from each city. Let $X$ and $Y$ be their respective incomes. Compute $P(X < Y)$.
|
$\frac{2}{5}$
|
0.4
|
5.3-7
|
stat
|
math
|
|||
Let $p$ equal the proportion of triathletes who suffered a training-related overuse injury during the past year. Out of 330 triathletes who responded to a survey, 167 indicated that they had suffered such an injury during the past year. Use these data to give a point estimate of $p$.
|
$0.5061$
|
0.5061
|
7.3-3
|
stat
|
math
|
|||
One characteristic of a car's storage console that is checked by the manufacturer is the time in seconds that it takes for the lower storage compartment door to open completely. A random sample of size $n=5$ yielded the following times:
$\begin{array}{lllll}1.1 & 0.9 & 1.4 & 1.1 & 1.0\end{array}$ Find the sample mean, $\bar{x}$.
|
$1.1$
|
1.1
|
6.1-1
|
stat
|
math
|
|||
Let $X_1, X_2, \ldots, X_{16}$ be a random sample from a normal distribution $N(77,25)$. Compute $P(74.2<\bar{X}<78.4)$.
|
$0.8561$
|
0.8561
|
5.5-1 (b)
|
stat
|
math
|
|||
Let $X_1$ and $X_2$ be independent random variables with probability density functions $f_1\left(x_1\right)=2 x_1, 0 < x_1 <1 $, and $f_2 \left(x_2\right) = 4x_2^3$ , $0 < x_2 < 1 $, respectively. Compute $P \left(0.5 < X_1 < 1\right.$ and $\left.0.4 < X_2 < 0.8\right)$.
|
$\frac{36}{125}$
|
1.44
|
5.3-3
|
stat
|
math
|
|||
If $X$ is a random variable with mean 33 and variance 16, use Chebyshev's inequality to find A lower bound for $P(23 < X < 43)$.
|
$0.84$
|
0.84
|
5.8-1 (a)
|
stat
|
math
|
|||
Let $Y_1 < Y_2 < \cdots < Y_8$ be the order statistics of eight independent observations from a continuous-type distribution with 70 th percentile $\pi_{0.7}=27.3$. Determine $P\left(Y_7<27.3\right)$.
|
$0.2553$
|
0.2553
|
6.3-5
|
stat
|
math
|
|||
Let $X$ and $Y$ be independent with distributions $N(5,16)$ and $N(6,9)$, respectively. Evaluate $P(X>Y)=$ $P(X-Y>0)$.
|
$0.4207$
|
0.4207
|
5.4-21
|
stat
|
math
|
|||
A quality engineer wanted to be $98 \%$ confident that the maximum error of the estimate of the mean strength, $\mu$, of the left hinge on a vanity cover molded by a machine is 0.25 . A preliminary sample of size $n=32$ parts yielded a sample mean of $\bar{x}=35.68$ and a standard deviation of $s=1.723$. How large a sample is required?
|
$257$
|
257
|
7.4-5
|
stat
|
math
|
|||
Let the distribution of $W$ be $F(8,4)$. Find the following: $F_{0.01}(8,4)$.
|
14.80
|
14.80
|
5.2-5
|
stat
|
math
|
|||
If the distribution of $Y$ is $b(n, 0.25)$, give a lower bound for $P(|Y / n-0.25|<0.05)$ when $n=500$.
|
$0.85$
|
0.85
|
5.8-5
|
stat
|
math
|
|||
Let $\bar{X}$ be the mean of a random sample of size 12 from the uniform distribution on the interval $(0,1)$. Approximate $P(1 / 2 \leq \bar{X} \leq 2 / 3)$.
|
$0.4772$
|
0.4772
|
5.6-1
|
stat
|
math
|
|||
Determine the constant $c$ such that $f(x)= c x^3(1-x)^6$, $0 < x < 1$ is a pdf.
|
840
|
840
|
5.2-9
|
stat
|
math
|
|||
Three drugs are being tested for use as the treatment of a certain disease. Let $p_1, p_2$, and $p_3$ represent the probabilities of success for the respective drugs. As three patients come in, each is given one of the drugs in a random order. After $n=10$ 'triples' and assuming independence, compute the probability that the maximum number of successes with one of the drugs exceeds eight if, in fact, $p_1=p_2=p_3=0.7$
|
$0.0384$
|
0.0384
|
5.3-15
|
stat
|
math
|
|||
Evaluate
$$
\int_0^{0.4} \frac{\Gamma(7)}{\Gamma(4) \Gamma(3)} y^3(1-y)^2 d y
$$ Using integration.
|
0.1792
|
0.1792
|
5.2-11
|
stat
|
math
|
|||
Let $X$ equal the maximal oxygen intake of a human on a treadmill, where the measurements are in milliliters of oxygen per minute per kilogram of weight. Assume that, for a particular population, the mean of $X$ is $\mu=$ 54.030 and the standard deviation is $\sigma=5.8$. Let $\bar{X}$ be the sample mean of a random sample of size $n=47$. Find $P(52.761 \leq \bar{X} \leq 54.453)$, approximately.
|
$0.6247$
|
0.6247
|
5.6-7
|
stat
|
math
|
|||
Two components operate in parallel in a device, so the device fails when and only when both components fail. The lifetimes, $X_1$ and $X_2$, of the respective components are independent and identically distributed with an exponential distribution with $\theta=2$. The cost of operating the device is $Z=2 Y_1+Y_2$, where $Y_1=\min \left(X_1, X_2\right)$ and $Y_2=\max \left(X_1, X_2\right)$. Compute $E(Z)$.
|
$5$
|
5
|
5.3-19
|
stat
|
math
|
|||
If $X$ is a random variable with mean 33 and variance 16, use Chebyshev's inequality to find An upper bound for $P(|X-33| \geq 14)$.
|
$0.082$
|
0.082
|
5.8-1
|
stat
|
math
|
|||
Suppose that the length of life in hours (say, $X$ ) of a light bulb manufactured by company $A$ is $N(800,14400)$ and the length of life in hours (say, $Y$ ) of a light bulb manufactured by company $B$ is $N(850,2500)$. One bulb is randomly selected from each company and is burned until 'death.' Find the probability that the length of life of the bulb from company $A$ exceeds the length of life of the bulb from company $B$ by at least 15 hours.
|
$0.3085$
|
0.3085
|
5.5-9 (a)
|
stat
|
math
|
|||
An urn contains 10 red and 10 white balls. The balls are drawn from the urn at random, one at a time. Find the probability that the fourth white ball is the fourth ball drawn if the sampling is done with replacement.
|
$\frac{1}{16}$
|
0.0625
|
Problem 1.4.15
|
stat
|
math
|
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