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# Suppose the retail of an automobile is $33,000 in 1999 and that it increases at 6% per year. a. Write the equation of the exponential function, in the form y=a(1+r)^t, that models the retail prrice of the automobile in 2014. b. Use the model to predict the retail price of the automobile in 2014. Exponential models ANSWERED asked 2020-12-07 Suppose the retail of an automobile is$33,000 in 1999 and that it increases at 6% per year.
a. Write the equation of the exponential function, in the form $$\displaystyle{y}={a}{\left({1}+{r}\right)}^{{t}}$$, that models the retail prrice of the automobile in 2014.
b. Use the model to predict the retail price of the automobile in 2014.

## Answers (1)

2020-12-08
a) The retail price of automobile is $33,000 in the year 1999. The price increases by 6% per year. The given exponential function is $$\displaystyle{y}={a}{\left({1}+{r}\right)}^{{t}}$$, where t is the years after 1999. Substitute the given values in the exponential function as follows. $$\displaystyle{P}{\left({t}\right)}={33},{000}{\left({1}+{6}\%\right)}^{{t}}$$ $$\displaystyle{P}{\left({t}\right)}={33},{000}{\left({1}+{0.06}\right)}^{{t}}$$ $$\displaystyle{P}{\left({t}\right)}={33},{000}{\left({1.06}\right)}^{{t}}$$ Thus, the function that models the retail price of automobile is $$\displaystyle{P}{\left({t}\right)}={33},{000}{\left({1.06}\right)}^{{t}}$$ Therefore, the correct answer is second option. b) From (a), the function that models the retail price of automobile is $$\displaystyle{P}{\left({t}\right)}={33},{000}{\left({1.06}\right)}^{{t}}$$ where t is the years after 1999. The given year is 2014 So, the value of t is 2014-1999=5. Evaluate the retail price of the automobile in 2004 as follows. $$\displaystyle{P}{\left({t}\right)}={33},{000}{\left({1.06}\right)}^{{5}}$$ $$\displaystyle{P}{\left({5}\right)}={33},{000}{\left({1.33822}\right)}$$ $$\displaystyle{P}{\left({5}\right)}={44161.4406}$$ $$\displaystyle{P}{\left({5}\right)}\approx{44},{161}$$ Therefore, the price of the automobile in the year 2004 is about$44,161.

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