if a quantity grows exponentially, the doubling time is the amount of time it takes the quantity to double, and is given by $(\ln \ 2) \text{/} k$

systems that exhibit exponential decay follow a model of the form $y = y_0 e^{\text{-} k t}$

systems that exhibit exponential growth follow a model of the form $y = y_0 e^{k t}$

if a quantity decays exponentially, the half-life is the amount of time it takes the quantity to be reduced by half. It is given by $(\ln \ 2) \text{/} k$

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Calculus Volume 1 · 6.8 Exponential Growth and Decay

Exponential Growth and Decay: practice problems

15 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

Exercises 349
True or False? If true, prove it. If false, find the true answer.
If you invest $$ 500,$ an annual rate of interest of $3 %$ yields more money in the first year than a $2.5 %$ continuous rate of interest.
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Exercises 351
True or False? If true, prove it. If false, find the true answer.
If given a half-life of t years, the constant $k$ for $y = e^{k t}$ is calculated by $k = ln (1 / 2) / t .$
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Exercises 353
For the following exercises, use $y = y_{0} e^{k t} .$
If bacteria increase by a factor of $10$ in $10$ hours, how many hours does it take to increase by $100 ?$
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Exercises 355
For the following exercises, use $y = y_{0} e^{k t} .$
If a relic contains $90 %$ as much radiocarbon as new material, can it have come from the time of Christ (approximately $2000$ years ago)? Note that the half-life of radiocarbon is $5730$ years.
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Exercises 357
For the following exercises, use $y = y_{0} e^{k t} .$
The populations of New York and Los Angeles are growing at $1 %$ and $1.4 %$ a year, respectively. Starting from $8$ million (New York) and $6$ million (Los Angeles), when are the populations equal? Round your answer to a whole number of years.
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Exercises 359
For the following exercises, use $y = y_{0} e^{k t} .$
The effect of advertising decays exponentially. If $40 %$ of the population remembers a new product after $3$ days, how long will $20 %$ remember it?
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Exercises 361
For the following exercises, use $y = y_{0} e^{k t} .$
If $y = 100$ at $t = 4$ and $y = 10$ at when does
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Exercises 363
For the following exercises, use $y = y_{0} e^{k t} .$
What continuous interest rate has the same yield as an annual rate of $9 % ?$
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Exercises 365
For the following exercises, use $y = y_{0} e^{k t} .$
You are trying to save $$ 50, 000$ in $20$ years for college tuition for your child. If interest is a continuous $10 %,$ how much do you need to invest initially?
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Exercises 367
For the following exercises, use $y = y_{0} e^{k t} .$
You are trying to thaw some vegetables that are at a temperature of $1 ° F .$ To thaw vegetables safely, you must put them in the refrigerator, which has an ambient temperature of $44 ° F .$ You check on your vegetables $2$ hours after putting them in the refrigerator to find that they are now Plot the resulting temperature curve and use it to determine when the vegetables reach
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Exercises 369
For the following exercises, use $y = y_{0} e^{k t} .$
The spent fuel of a nuclear reactor contains plutonium-239, which has a half-life of $24, 000$ years. If $1$ barrel containing $10 kg$ of plutonium-239 is sealed, how many years must pass until only $10 g$ of plutonium-239 is left?
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Exercises 370
For the next set of exercises, use the following table, which features the world population by decade.
Source: http://www.factmonster.com/ipka/A0762181.html.
Years since 1950 Population (millions)
$0$ $2, 556$
$10$ $3, 039$
$20$ $3, 706$
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Exercises 371
For the next set of exercises, use the following table, which features the world population by decade.
Source: http://www.factmonster.com/ipka/A0762181.html.
Years since 1950 Population (millions)
$0$ $2, 556$
$10$ $3, 039$
$20$ $3, 706$
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Exercises 373
For the next set of exercises, use the following table, which features the world population by decade.
Source: http://www.factmonster.com/ipka/A0762181.html.
Years since 1950 Population (millions)
$0$ $2, 556$
$10$ $3, 039$
$20$ $3, 706$
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Exercises 375
For the next set of exercises, use the following table, which shows the population of San Francisco during the 19th century.
Source: http://www.sfgenealogy.com/sf/history/hgpop.htm.
Years since 1850 Population (thousands)
$0$ $21.00$
$10$ $56.80$
$20$ $149.5$
$30$
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Key terms · 4

doubling time: if a quantity grows exponentially, the doubling time is the amount of time it takes the quantity to double, and is given by $(\ln \ 2) \text{/} k$
exponential decay: systems that exhibit exponential decay follow a model of the form $y = y_0 e^{\text{-} k t}$
exponential growth: systems that exhibit exponential growth follow a model of the form $y = y_0 e^{k t}$
half-life: if a quantity decays exponentially, the half-life is the amount of time it takes the quantity to be reduced by half. It is given by $(\ln \ 2) \text{/} k$

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Exponential Growth and Decay: practice problems

15 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

Exercises 349

True or False? If true, prove it. If false, find the true answer.

If you invest $$ 500,$ an annual rate of interest of $3 %$ yields more money in the first year than a $2.5 %$ continuous rate of interest.

Exercises 351

True or False? If true, prove it. If false, find the true answer.

If given a half-life of t years, the constant $k$ for $y = e^{k t}$ is calculated by $k = ln (1 / 2) / t .$

Exercises 353