# 4.7e: Exercises - Exponential Applications

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### A: Concepts

Exercise $$\PageIndex{A}$$

1) With what kind of exponential model would half-life be associated? What role does half-life play in these models?

2) What is carbon dating? Why does it work? Give an example in which carbon dating would be useful.

3) With what kind of exponential model would doubling time be associated? What role does doubling time play in these models?

4) Define Newton’s Law of Cooling. Then name at least three real-world situations where Newton’s Law of Cooling would be applied.

5) What is an order of magnitude? Why are orders of magnitude useful? Give an example to explain.

7) A doctor and injects a patient with $$13$$ milligrams of radioactive dye that decays exponentially. After $$12$$ minutes, there are $$4.75$$ milligrams of dye remaining in the patient’s system. Which is an appropriate model for this situation?

a.  $$f(t)=13(0.0805)^t$$ $$\quad$$ b.  $$f(t)=13e^{0.9195t}$$ $$\quad$$ c.  $$f(t)=13e^{(-0.0839t)}$$ $$\quad$$ d.  $$f(t)=\frac{4.75}{1+13e^{-0.83925t}}$$

9) What situations are best modeled by a logistic equation? Give an example, and state a case for why the example is a good fit.

10) What is a carrying capacity? What kind of model has a carrying capacity built into its formula? Why does this make sense?

1. Half-life is a measure of decay and is thus associated with exponential decay models. The half-life of a substance or quantity is the amount of time it takes for half of the initial amount of that substance or quantity to decay.

3. Doubling time is a measure of growth and is thus associated with exponential growth models. The doubling time of a substance or quantity is the amount of time it takes for the initial amount of that substance or quantity to double in size.

5. An order of magnitude is the nearest power of ten by which a quantity exponentially grows. It is also an approximate position on a logarithmic scale; Sample response: Orders of magnitude are useful when making comparisons between numbers that differ by a great amount. For example, the mass of Saturn is $$95$$ times greater than the mass of Earth. This is the same as saying that the mass of Saturn is about $$10^2$$ times, or $$2$$ orders of magnitude greater, than the mass of Earth.

7. C

9. Logistic models are best used for situations that have limited values. For example, populations cannot grow indefinitely since resources such as food, water, and space are limited, so a logistic model best describes populations.

### B: Compound Interest

Exercise $$\PageIndex{B}$$

Find current amount

1. Billy’s grandfather invested in a savings bond that earned $$5.5$$% annual interest that was compounded annually. Currently, $$30$$ years later, the savings bond is valued at $$$10,000$$. Determine what the initial investment was. 2. Given a yearly interest rate of 3.5% and an initial principle of$100,000, find the amount $$A$$ accumulated in 5 years for interest that is compounded a. daily, b., monthly, c. quarterly, and d. yearly.
3. Starting with $100,000 invested at an annual interest rate of $$5.5$$% compounded continuously, find the amount accumulated after 5 years. 4. In 1935 Frank opened an account earning $$3.8$$% annual interest that was compounded quarterly. He rediscovered this account while cleaning out his garage in 2005. If the account is now worth$$$11,294.30$$, how much was his initial deposit in 1935?

Find principal

1. Given that the bank is offering $$4.2$$% annual interest compounded monthly, what principal is needed to earn $$$25,000$$ in interest for one year? 2. Given that the bank is offering $$3.5$$% annual interest compounded continuously, what principal is needed to earn$$$12,000$$ in interest for one year?

Find interest rate

1. Find the annual interest rate at which an account earning continuously compounding interest has a doubling time of $$9$$ years.
2. Find the annual interest rate at which an account earning interest that is compounded monthly has a doubling time of $$10$$ years.
3. Alice invests $$$15,000$$ at age $$30$$ from the signing bonus of her new job. She hopes the investments will be worth$$$30,000$$ when she turns $$40$$. If the interest compounds continuously, approximately what rate of growth will she need to achieve her goal?
4. Sung Lee invests $$$5,000$$ at age $$18$$. He hopes the investments will be worth$$$10,000$$ when he turns $$25$$. If the interest compounds continuously, approximately what rate of growth will he need to achieve his goal? Is that a reasonable expectation?

Find time

1. Jill invested $$$1,450$$ in an account earning $$4 \frac{5}{8}$$% annual interest that is compounded monthly. 1. How much will be in the account after $$6$$ years? 2. How long will it take the account to grow to$$$2,200$$?
2. James invested $$$825$$ in an account earning $$5 \frac{2}{5}$$% annual interest that is compounded monthly. 1. How much will be in the account after $$4$$ years? 2. How long will it take the account to grow to$$$1,500$$?
3. Bill wants to grow his $$$75,000$$ inheritance to$$$100,000$$ before spending any of it. How long will this take if the bank is offering $$5.2$$% annual interest compounded quarterly?
4. Mary needs $$$25,000$$ for a down payment on a new home. If she invests her savings of$$$21,350$$  in an account earning $$4.6$$% annual interest that is compounded semi-annually, how long will it take to grow to the amount that she needs?
5. Joe invested his $$$8,700$$ savings in an account earning $$6 \frac{3}{4}$$% annual interest that is compounded continuously. How long will it take to earn$$$300$$ in interest?
6. Miriam invested $$$12,800$$ in an account earning $$5 \frac{1}{4}$$% annual interest that is compounded monthly. How long will it take to earn$$$1,200$$ in interest?
7. Raul invested $$$8,500$$ in an online money market fund earning $$4.8$$% annual interest that is compounded continuously. 1. How much will be in the account after $$2$$ years? 2. How long will it take the account to grow to$$$10,000$$?
8. Ian deposited $$$500$$ in an account earning $$3.9$$% annual interest that is compounded continuously. 1. How much will be in the account after $$3$$ years? 2. How long will it take the account to grow to$$$1,500$$?

Find doubling time

1. Jose invested his $$$3,500$$ bonus in an account earning $$5 \frac{1}{2}$$% annual interest that is compounded quarterly. How long will it take to double his investment? 2. Maria invested her$$$4,200$$ savings in an account earning $$6 \frac{3}{4}$$% annual interest that is compounded semi-annually. How long will it take to double her savings?
3. Calculate the doubling time of an investment made at $$7$$% annual interest that is compounded: (a) monthly (b) continuously
4. Coralee invests $$$5,000$$ in an account that compounds interest monthly and earns $$7$$%. How long will it take for her money to double? 5. If money is invested in an account earning $$3.85$$% annual interest that is compounded continuously, how long will it take the amount to double? 6. Calculate the doubling time of an investment that is earning continuously compounding interest at an annual interest rate of: (a) $$4$$% (b) $$6$$% Find Tripling time 1. Jill invested$$$1,450$$ in an account earning $$4 \frac{5}{8}$$% annual interest that is compounded monthly.
1. Find the amount accumulated after 5 years and 10 years.
2. Determine how long it takes for the original investment to triple.
2. Alice invested her savings of $$$7,000$$ in an account earning $$4.5$$% annual interest that is compounded monthly. How long will it take the account to triple in value? 3. Mary invested her$$$42,000$$ bonus in an account earning $$7.2$$% annual interest that is compounded continuously. How long will it take the account to triple in value?

### F: Newton' s Law of Heating and Cooling

Exercise $$\PageIndex{F}$$

111. The temperature of an object in degrees Fahrenheit after $$t$$ minutes is represented by the equation $$T(t)=68e^{-0.0174t}+72$$ .   To the nearest degree, what is the temperature of the object after one and a half hours?

112. A pot of boiling soup with an internal temperature of $$100^{\circ}$$ Fahrenheit was taken off the stove to cool in a $$69^{\circ}$$ F room. After fifteen minutes, the internal temperature of the soup was $$95^{\circ}$$ F.

a) Use Newton’s Law of Cooling to write a formula that models this situation.

b) To the nearest minute, how long will it take the soup to cool to $$80^{\circ}$$ F?

c) To the nearest degree, what will the temperature be after $$2$$ and a half hours?

113. A turkey is taken out of the oven with an internal temperature of $$165^{\circ}$$ F and is allowed to cool in a $$75^{\circ}$$ F room. After half an hour, the internal temperature of the turkey is $$145^{\circ}$$ F.

(a) Write a formula that models this situation.

(b) To the nearest degree, what will the temperature be after $$50$$ minutes?

(c) To the nearest minute, how long will it take the turkey to cool to $$110^{\circ}$$ F?

111. about $$86.2^{\circ}$$ F     113. (a) $$T(t) = 90e^{(-0.008377t)}+75$$, where $$t$$ is in minutes.  (b) $$134^\circ$$ F  (c) about $$113$$ minutes

### G: Logistic Growth

Exercise $$\PageIndex{G}$$ 121. The population of a fish farm in $$t$$ years is modeled by the equation $$P(t)=\dfrac{1000}{1+9e^{-0.6t}}$$. The graph of the function is at the right.

a) What is the initial population of fish?

b) To the nearest tenth, what is the doubling time for the fish population?

c) To the nearest whole number, what will the fish population be after $$2$$ years?

d) To the nearest tenth, how long will it take for the population to reach $$900$$?

e) What is the carrying capacity for the fish population? Justify your answer using the graph of $$P$$.

122. The equation $$N(t)=\dfrac{500}{1+49e^{-0.7t}}$$ models the number of people in a town who have heard a rumor after $$t$$ days.

a) How many people started the rumor?

b) To the nearest whole number, how many people will have heard the rumor after $$3$$ days?

c) As $$t$$ increases without bound, what value does $$N(t)$$ approach? Interpret your answer.

123. What is the $$y$$-intercept of the logistic growth model $$y=\dfrac{c}{1+ae^{-rx}}$$ ?   Show the steps for calculation. What does this point tell us about the population?

124. A logistic model is given by the equation $$P(t)=\dfrac{90}{1+5e^{-0.42t}}$$. To the nearest hundredth, for what value of $$t$$ does $$P(t)=45$$

125. To the nearest whole number, what is the initial value of a population modeled by the logistic equation $$P(t)=\dfrac{175}{1+6.995e^{-0.68t}}$$? What is the carrying capacity?151) What is the $$y$$-intercept on the graph of the logistic model given in the previous exercise?

126. The number of cells in a certain bacteria sample is approximated by the logistic growth model $$N(t)=\frac{1.2 \times 10^{5}}{1+9 e^{-0.32t}}$$, where $$t$$ represents time in hours. Determine the time it takes the sample to grow to $$24,000$$ cells.

127. Given the logistic growth model

a) Find and interpret $$f(0)$$$.$Round to the nearest tenth.

b) Find and interpret $$f(4)$$ .   Round to the nearest tenth.

c) Find the carrying capacity.

128. The population $$P$$ of an endangered species habitat for wolves is modeled by the function $$P(x)=\dfrac{558}{1+54.8e^{-0.462x}}$$where $$x$$ is given in years.

a) What was the initial population of wolves transported to the habitat?

b) How many wolves will the habitat have after $$3$$ years?

c) How many years will it take before there are $$100$$ wolves in the habitat?

121. (a) 100 fish,   (b) about $$1.4$$ years,     (c) $$269$$ fish,     (d) about $$7.3$$ years,     (e) 1000 fish - the upper horizontal asymptote bound
123. $$\dfrac{c}{1+a}$$;  initial population size          125. $$P(0)=22$$; $$175$$
127. (a) $$f(0)\approx 16.7$$; The amount initially present is about $$16.7$$ units.       (b) The quantity after 4 hours is $$149.6$$ units      (c)  $$150$$
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