3.9E: Excercises

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Exercise $\PageIndex{1}$ : Growth and Decay

1. The half-life of a radioactive substance is 3200 years. Find the quantity $Q(t)$ of the substance left at time $t > 0$ if $Q(0)=20$ g.

2. The half-life of a radioactive substance is 2 days. Find the time required for a given amount of the material to decay to 1/10 of its original mass.

3. A radioactive material loses 25% of its mass in 10 minutes. What is its half-life?

4. A tree contains a known percentage $p_0$ of a radioactive substance with half-life $\tau$. When the tree dies the substance decays and isn’t replaced. If the percentage of the substance in the fossilized remains of such a tree is found to be $p_1$, how long has the tree been dead?

5. If $t_p$ and $t_q$ are the times required for a radioactive material to decay to $1/p$ and $1/q$ times its original mass (respectively), how are $t_p$ and $t_q$ related?

6. Find the decay constant $k$ for a radioactive substance, given that the mass of the substance is $Q_1$ at time $t_1$ and $Q_2$ at time $t_2$.

7. A process creates a radioactive substance at the rate of 2 g/hr and the substance decays at a rate proportional to its mass, with constant of proportionality $k=.1 (\mbox{hr})^{-1}$. If $Q(t)$ is the mass of the substance at time $t$, find $\lim_{t\to\infty}Q(t)$.

8. A bank pays interest continuously at the rate of 6%. How long does it take for a deposit of $Q_0$ to grow in value to $2Q_0$?

9. At what rate of interest, compounded continuously, will a bank deposit double in value in 8 years?

10. A savings account pays 5% per annum interest compounded continuously. The initial deposit is $Q_0$ dollars. Assume that there are no subsequent withdrawals or deposits.

How long will it take for the value of the account to triple?
What is $Q_0$ if the value of the account after 10 years is $100,000 dollars?

11. A candymaker makes 500 pounds of candy per week, while his large family eats the candy at a rate equal to $Q(t)/10$ pounds per week, where $Q(t)$ is the amount of candy present at time $t$.

Find $Q(t)$ for $t > 0$ if the candymaker has 250 pounds of candy at $t=0$.
Find $\lim_{t\to\infty} Q(t)$.

12. Suppose a substance decays at a yearly rate equal to half the square of the mass of the substance present. If we start with 50 g of the substance, how long will it be until only 25 g remain?

13. A super bread dough increases in volume at a rate proportional to the volume $V$ present. If $V$ increases by a factor of 10 in 2 hours and $V(0)=V_0$, find $V$ at any time $t$. How long will it take for $V$ to increase to $100 V_0$?

14. A radioactive substance decays at a rate proportional to the amount present, and half the original quantity $Q_0$ is left after 1500 years. In how many years would the original amount be reduced to $3Q_0/4$? How much will be left after 2000 years?

15. A wizard creates gold continuously at the rate of 1 ounce per hour, but an assistant steals it continuously at the rate of 5% of however much is there per hour. Let $W(t)$ be the number of ounces that the wizard has at time $t$. Find $W(t)$ and $\lim_{t\to\infty}W(t)$ if $W(0)=1$.

16. A process creates a radioactive substance at the rate of 1 g/hr, and the substance decays at an hourly rate equal to 1/10 of the mass present (expressed in grams). Assuming that there are initially 20 g, find the mass $S(t)$ of the substance present at time $t$, and find $\lim_{t\to\infty} S(t)$.

17. A tank is empty at $t=0$. Water is added to the tank at the rate of 10 gal/min, but it leaks out at a rate (in gallons per minute) equal to the number of gallons in the tank. What is the smallest capacity the tank can have if this process is to continue forever?

18. A person deposits $25,000 in a bank that pays 5% per year interest, compounded continuously. The person continuously withdraws from the account at the rate of $750 per year. Find $V(t)$, the value of the account at time $t$ after the initial deposit.

19. A person has a fortune that grows at rate proportional to the square root of its worth. Find the worth $W$ of the fortune as a function of $t$ if it was $1 million 6 months ago and is $4 million today.

20. Let $p=p(t)$ be the quantity of a product present at time $t$. The product is manufactured continuously at a rate proportional to $p$, with proportionality constant 1/2, and it is consumed continuously at a rate proportional to $p^2$, with proportionality constant 1/8. Find $p(t)$ if $p(0)=100$.

21.

a. In the situation of Example 4.1.6 find the exact value P(t) of the person’s account after t years, where t is an integer. Assume that each year has exactly 52 weeks, and include the year-end deposit in the computation.

HINT: At time t the initial $1000 has been on deposit for $t$ years. There have been $52t$ deposits of $$50$ each. The first $$50$ has been on deposit for $t − 1/52$ years, the second for $t − 2/52$ years ... in general, the j th $$50$ has been on deposit for $t − j/52$ years ($1 ≤ j ≤ 52t$). Find the present value of each $$50$ deposit assuming $6$% interest compounded continuously, and use the formula \[1+x+x^{2}+ . . . + x^{n}=\frac{1-x^{n+1}}{1-x}(x\neq 1)\] to find their total value.

b. Let

\[p(t)={Q(t)-P(t)\over P(t)}\]

be the relative error after $t$ years. Find

\[p(\infty)=\lim_{t\to\infty}p(t).\]

22. A homebuyer borrows $P_0$ dollars at an annual interest rate $r$, agreeing to repay the loan with equal monthly payments of $M$ dollars per month over $N$ years.

a. Derive a differential equation for the loan principal (amount that the homebuyer owes) $P(t)$ at time $t>0$, making the simplifying assumption that the homebuyer repays the loan continuously rather than in discrete steps. (See Example 4.1.6.)

b. Solve the equation derived in (a).

c. Use the result of (b) to determine an approximate value for $M$ assuming that each year has exactly 12 months of equal length.

d. It can be shown that the exact value of $M$ is given by

\[M={rP_0\over 12}\left(1-(1+r/12)^{-12N}\right)^{-1}.\]

Compare the value of $M$ obtained from the answer in (c) to the exact value if (i) $P_0=\$50,000$, $r=7{1\over2}$%, $N=20$ (ii) $P_0=\$150,000$, $r=9.0$%, $N=30$.

23. Assume that the homebuyer of Exercise 4.1.22 elects to repay the loan continuously at the rate of $\alpha M$ dollars per month, where $\alpha$ is a constant greater than 1. (This is called accelerated payment.)

Determine the time $T(\alpha)$ when the loan will be paid off and the amount $S(\alpha)$ that the homebuyer will save.
Suppose $P_0=\$50,000$, $r=8$%, and $N=15$. Compute the savings realized by accelerated payments with $\alpha=1.05,1.10$, and $1.15$.

24. A benefactor wishes to establish a trust fund to pay a researcher’s salary for $T$ years. The salary is to start at $S_0$ dollars per year and increase at a fractional rate of $a$ per year. Find the amount of money $P_0$ that the benefactor must deposit in a trust fund paying interest at a rate $r$ per year. Assume that the researcher’s salary is paid continuously, the interest is compounded continuously, and the salary increases are granted continuously.

25. A radioactive substance with decay constant $k$ is produced at the rate of

\[{at\over1+btQ(t)}\]

units of mass per unit time, where $a$ and $b$ are positive constants and $Q(t)$ is the mass of the substance present at time $t$; thus, the rate of production is small at the start and tends to slow when $Q$ is large.

Set up a differential equation for $Q$.
Choose your own positive values for $a$, $b$, $k$, and $Q_0=Q(0)$. Use a numerical method to discover what happens to $Q(t)$ as $t\to\infty$. (Be precise, expressing your conclusions in terms of $a$, $b$, $k$. However, no proof is required.)

26. Follow the instructions of Exercise 4.1.25, assuming that the substance is produced at the rate of $at/(1+bt(Q(t))^2)$ units of mass per unit of time.

27. Follow the instructions of Exercise 4.1.25, assuming that the substance is produced at the rate of $at/(1+bt)$ units of mass per unit of time.

Exercise $\PageIndex{2}$: Cooling and Mixing

1. A thermometer is moved from a room where the temperature is $70^\circ$F to a freezer where the temperature is $12^\circ F$. After $30$ seconds the thermometer reads $40^\circ$F. What does it read after $2$ minutes?

2. A fluid initially at $100^\circ$C is placed outside on a day when the temperature is $-10^\circ$C, and the temperature of the fluid drops $20^\circ$C in one minute. Find the temperature $T(t)$ of the fluid for $t > 0$.

3. At 12:00 pm a thermometer reading $10^\circ$F is placed in a room where the temperature is $70^\circ$F. It reads $56^\circ$ when it is placed outside, where the temperature is $5^\circ$F, at 12:03. What does it read at 12:05 pm?

4. A thermometer initially reading $212^\circ$F is placed in a room where the temperature is $70^\circ$F. After 2 minutes the thermometer reads $125^\circ$F.

What does the thermometer read after $4$ minutes?
When will the thermometer read $72^\circ$F?
When will the thermometer read $69^\circ$F?

5. An object with initial temperature $150^\circ$C is placed outside, where the temperature is $35^\circ$C. Its temperatures at 12:15 and 12:20 are $120^\circ$C and $90^\circ$C, respectively.

At what time was the object placed outside?
When will its temperature be $40^\circ$C?

6. An object is placed in a room where the temperature is $20^\circ$C. The temperature of the object drops by $5^\circ$C in $4$ minutes and by $7^\circ$C in $8$ minutes. What was the temperature of the object when it was initially placed in the room?

7. A cup of boiling water is placed outside at 1:00 pm. One minute later the temperature of the water is $152^\circ$F. After another minute its temperature is $112^\circ$F. What is the outside temperature?

8. A tank initially contains $40$ gallons of pure water. A solution with $1$ gram of salt per gallon of water is added to the tank at $3$ gal/min, and the resulting solution drains out at the same rate. Find the quantity $Q(t)$ of salt in the tank at time $t > 0$.

9. A tank initially contains a solution of $10$ pounds of salt in $60$ gallons of water. Water with $1/2$ pound of salt per gallon is added to the tank at $6$ gal/min, and the resulting solution leaves at the same rate. Find the quantity $Q(t)$ of salt in the tank at time $t > 0$.

10. A tank initially contains $100$ liters of a salt solution with a concentration of $.1$ g/liter. A solution with a salt concentration of $.3$ g/liter is added to the tank at $5$ liters/min, and the resulting mixture is drained out at the same rate. Find the concentration $K(t)$ of salt in the tank as a function of $t$.

11. A $200$ gallon tank initially contains $100$ gallons of water with $20$ pounds of salt. A salt solution with $1/4$ pound of salt per gallon is added to the tank at $4$ gal/min, and the resulting mixture is drained out at $2$ gal/min. Find the quantity of salt in the tank as it is about to overflow.

12. Suppose water is added to a tank at 10 gal/min, but leaks out at the rate of $1/5$ gal/min for each gallon in the tank. What is the smallest capacity the tank can have if the process is to continue indefinitely?

13. A chemical reaction in a laboratory with volume $V$ (in ft$^3$) produces $q_1$ ft$^3$/min of a noxious gas as a byproduct. The gas is dangerous at concentrations greater than $\overline c$, but harmless at concentrations $\le \overline c$. Intake fans at one end of the laboratory pull in fresh air at the rate of $q_2$ ft$^3$/min and exhaust fans at the other end exhaust the mixture of gas and air from the laboratory at the same rate. Assuming that the gas is always uniformly distributed in the room and its initial concentration $c_0$ is at a safe level, find the smallest value of $q_2$ required to maintain safe conditions in the laboratory for all time.

14. A $1200$-gallon tank initially contains $40$ pounds of salt dissolved in $600$ gallons of water. Starting at $t_0=0$, water that contains $1/2$ pound of salt per gallon is added to the tank at the rate of $6$ gal/min and the resulting mixture is drained from the tank at $4$ gal/min. Find the quantity $Q(t)$ of salt in the tank at any time $t > 0$ prior to overflow.

15. Tank $T_1$ initially contain $50$ gallons of pure water. Starting at $t_0=0$, water that contains $1$ pound of salt per gallon is poured into $T_1$ at the rate of $2$ gal/min. The mixture is drained from $T_1$ at the same rate into a second tank $T_2$, which initially contains $50$ gallons of pure water. Also starting at $t_0=0$, a mixture from another source that contains $2$ pounds of salt per gallon is poured into $T_2$ at the rate of $2$ gal/min. The mixture is drained from $T_2$ at the rate of $4$ gal/min.

Find a differential equation for the quantity $Q(t)$ of salt in tank $T_2$ at time $t > 0$.
Solve the equation derived in (a) to determine $Q(t)$.
Find $\lim_{t\to\infty}Q(t)$.

16. Suppose an object with initial temperature $T_0$ is placed in a sealed container, which is in turn placed in a medium with temperature $T_m$. Let the initial temperature of the container be $S_0$. Assume that the temperature of the object does not affect the temperature of the container, which in turn does not affect the temperature of the medium. (These assumptions are reasonable, for example, if the object is a cup of coffee, the container is a house, and the medium is the atmosphere.)

Assuming that the container and the medium have distinct temperature decay constants $k$ and $k_m$ respectively, use Newton’s law of cooling to find the temperatures $S(t)$ and $T(t)$ of the container and object at time $t$.
Assuming that the container and the medium have the same temperature decay constant $k$, use Newton’s law of cooling to find the temperatures $S(t)$ and $T(t)$ of the container and object at time $t$.
Find $\lim._{t\to\infty}S(t)$ and $\lim_{t\to\infty}T(t)$.

17. In our previous examples and exercises concerning Newton’s law of cooling we assumed that the temperature of the medium remains constant. This model is adequate if the heat lost or gained by the object is insignificant compared to the heat required to cause an appreciable change in the temperature of the medium. If this isn’t so, we must use a model that accounts for the heat exchanged between the object and the medium. Let $T=T(t)$ and $T_m=T_m(t)$ be the temperatures of the object and the medium, respectively, and let $T_0$ and $T_{m0}$ be their initial values. Again, we assume that $T$ and $T_m$ are related by Newton’s law of cooling,

\[T'=-k(T-T_m). \tag{A}\]

We also assume that the change in heat of the object as its temperature changes from $T_0$ to $T$ is $a(T-T_0)$ and that the change in heat of the medium as its temperature changes from $T_{m0}$ to $T_m$ is $a_m(T_m-T_{m0})$, where $a$ and $a_m$ are positive constants depending upon the masses and thermal properties of the object and medium, respectively. If we assume that the total heat of the system consisting of the object and the medium remains constant (that is, energy is conserved), then

\[a(T-T_0)+a_m(T_m-T_{m0})=0. \tag{B}\]

Equation (A) involves two unknown functions $T$ and $T_m$. Use (A) and (B) to derive a differential equation involving only $T$.
Find $T(t)$ and $T_m(t)$ for $t>0$.
Find $\lim_{t\to\infty}T(t)$ and $\lim_{t\to\infty}T_m(t)$.

18. Control mechanisms allow fluid to flow into a tank at a rate proportional to the volume $V$ of fluid in the tank, and to flow out at a rate proportional to $V^2$. Suppose $V(0)=V_0$ and the constants of proportionality are $a$ and $b$, respectively. Find $V(t)$ for $t>0$ and find $\lim_{t\to\infty}V(t)$.

19. Identical tanks $T_1$ and $T_2$ initially contain $W$ gallons each of pure water. Starting at $t_0=0$, a salt solution with constant concentration $c$ is pumped into $T_1$ at $r$ gal/min and drained from $T_1$ into $T_2$ at the same rate. The resulting mixture in $T_2$ is also drained at the same rate. Find the concentrations $c_1(t)$ and $c_2(t)$ in tanks $T_1$ and $T_2$ for $t>0$.

20. An infinite sequence of identical tanks $T_1$, $T_2$, …, $T_n$, …, initially contain $W$ gallons each of pure water. They are hooked together so that fluid drains from $T_n$ into $T_{n+1}\,(n=1,2,\cdots)$. A salt solution is circulated through the tanks so that it enters and leaves each tank at the constant rate of $r$ gal/min. The solution has a concentration of $c$ pounds of salt per gallon when it enters $T_1$.

Find the concentration $c_n(t)$ in tank $T_n$ for $t>0$.
Find $\lim_{t\to\infty}c_n(t)$ for each $n$.

21. Tanks $T_1$ and $T_2$ have capacities $W_1$ and $W_2$ liters, respectively. Initially they are both full of dye solutions with concentrations $c_{1}$ and $c_2$ grams per liter. Starting at $t_0=0$, the solution from $T_1$ is pumped into $T_2$ at a rate of $r$ liters per minute, and the solution from $T_2$ is pumped into $T_1$ at the same rate.

Find the concentrations $c_1(t)$ and $c_2(t)$ of the dye in $T_1$ and $T_2$ for $t>0$.
Find $\lim_{t\to\infty}c_1(t)$ and $\lim_{t\to\infty}c_2(t)$.

22. Consider the mixing problem of Example 4.2.3, but without the assumption that the mixture is stirred instantly so that the salt is always uniformly distributed throughout the mixture. Assume instead that the distribution approaches uniformity as $t\to\infty$. In this case the differential equation for $Q$ is of the form

\[Q'+{a(t)\over150}Q=2\]

where $\lim_{t\to\infty}a(t)=1$.

Assuming that $Q(0)=Q_0$, can you guess the value of $\lim_{t\to\infty}Q(t)$?.
Use numerical methods to confirm your guess in the these cases:

\[\text{(i) } a(t)=t/(1+t) \quad \text{(ii) } a(t)=1-e^{-t^2} \quad \text{(iii) } a(t)=1-\sin(e^{-t}).\]

23. Consider the mixing problem of Example 4.2.4 in a tank with infinite capacity, but without the assumption that the mixture is stirred instantly so that the salt is always uniformly distributed throughout the mixture. Assume instead that the distribution approaches uniformity as $t\to\infty$. In this case the differential equation for $Q$ is of the form

\[Q'+{a(t)\over t+100}Q=1\]

where $\lim_{t\to\infty}a(t)=1$.

Let $K(t)$ be the concentration of salt at time $t$. Assuming that $Q(0)=Q_0$, can you guess the value of $\lim_{t\to\infty}K(t)$?
Use numerical methods to confirm your guess in the these cases:

\[\text{(i) } a(t)=t/(1+t)\quad \text{(ii) } a(t)=1-e^{-t^2} \quad \text{(iii) } a(t)=1+\sin(e^{-t}).\]