Question

# 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.

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.