Exponential Modeling

Section 2.2 Exponential Modeling

Exponential functions are used to model situations in which the rate of growth of an quantity is proportional to the quantity’s size. Exponential models are used in situations such as money earning compound interest, population growth, and radioactive decay.

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Goals:

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E: Be able to solve an equation/inequality with an unknown exponent.
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E: Be able to model a situation with appropriate exponential equation(s) and interpret the solution.
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E: Be able to determine the equation of an exponential function given a table of values.
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F: Be able to compute the average rate of change of a given function on a given interval.
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Example 2.2.1.

Pam opens a banking account with $500. The account earns 1.5% compounded annually. Let $t$ be the number of years the bank account has been open, and $B(t)$ the balance in the account.

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Make a table showing the balance, $B(t)\text{,}$ in the account at $t=0\text{,}$ 1, 2, 3, and 4 years.
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Write a formula for the function $B\text{.}$
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Use your formula for $B$ to determine the balance in 20 years.
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When will the balance in the account reach $800?
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Problem 2.2.2.

Tyus opens a banking account with $800. The account earns 3% compounded annually. Let $t$ be the number of years the bank account has been open, and $A(t)$ the balance in the account.

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Make a table showing the balance, $A(t)\text{,}$ in the account at $t=0\text{,}$ 1, 2, 3, and 4 years.
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Write a formula for the function $A\text{.}$
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Use your formula for the function $A$ to determine the balance in 20 years.
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When will the balance in the account reach $1400?
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Problem 2.2.3.

The bacteria in a dish have an initial population of 1000, and are growing such that the population doubles every 45 minutes. Let $t$ be the number of minutes that have passed since the initial population was measured, and let $P(t)$ be the population at time $t$ minutes.

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Make a table showing the population, $P(t)\text{,}$ at times $t=0\text{,}$ 45, 90, and 135 minutes.
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Write an equation for the function $P\text{.}$
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Use your function equation to determine the population of bacteria in 6 hours (360 minutes).
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When will there be 250,000 bacteria?
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Determine the average number of bacteria added per hour in the first 6 hours.
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Problem 2.2.4.

A Christmas tree lot sold 200 trees in 2010 and 350 trees in 2015.

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Assuming the number of trees sold is increasing exponentially, write a formula for the number of trees sold $t$ years after 2010.
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If the number of trees sold continues to increase exponentially, how many trees will be sold in 2019?
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If the number of trees sold continues to increase exponentially, in what year will 500 trees be sold?
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Problem 2.2.5.

The population of Dry Gulch has been decreasing by $\frac{1}{2}$ every 20 years. In 1910 the population of Dry Gulch was 3800 people.

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Write a formula for the population of Dry Gulch $t$ years after 1910.
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If the number of people in Dry Gulch continued to decrease exponentially, how many people were there in the town in 2010?
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If the number of people in Dry Gulch continues to decrease exponentially, when will the town become a ghost town (a population of less than 1 person)?
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Problem 2.2.6.

A camera costs $110 now. The cost of the camera increases by 6% annually.

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Write a formula for the cost of the camera $t$ years from now.
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If the cost continues to rise exponentially, how much will the camera cost in 3 years?
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If the cost continues to rise exponentially, how long will it take the cost of the camera to reach $200?
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Problem 2.2.7.

Carbon dating is used to determine the age of bones, tools and other relics. Carbon-14 has a half life of $5728$ years, meaning that if an object is found to have $50$% of its original carbon, it is $5728$ years old.

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Make a table for the amount of Carbon-14, $C(t)\text{,}$ at time $t=0\text{,}$ $5728\text{,}$ and $11456$ years, supposing that the initial amount of carbon is an unknown $a\text{.}$
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Write an equation for the amount of Carbon-14, $C(t)\text{,}$ at time $t\text{.}$
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In 1990 a body was found in the Sierra Nevada mountain range. An examination of the tissue found that 27% of the carbon-14 present at the time of death had decayed. How long ago did the man die?
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