the depreciation rate r for a new car is given by the function $r = 1-left(\frac{v}{35400}\right)^{1/4}$…

the depreciation rate r for a new car is given by the function $r = 1-left(\frac{v}{35400}\right)^{1/4}$, where v is the value of the car when it is four years old. a. what is the depreciation rate for a $35,400 luxury car that is worth $18,400 after 4 years? b. write v as a function of r. a. what is the depreciation rate for a $35,400 luxury car that is worth $18,400 after 4 years? % (round to the nearest tenth as needed.)

the depreciation rate r for a new car is given by the function $r = 1-left(\frac{v}{35400}\right)^{1/4}$, where v is the value of the car when it is four years old. a. what is the depreciation rate for a $35,400 luxury car that is worth $18,400 after 4 years? b. write v as a function of r. a. what is the depreciation rate for a $35,400 luxury car that is worth $18,400 after 4 years? % (round to the nearest tenth as needed.)

Answer

Explanation:

Step1: Substitute values into formula

Given $V = 18400$ and the initial value is $35400$, substitute into $r = 1-\left(\frac{V}{35400}\right)^{\frac{1}{4}}$. So $r = 1-\left(\frac{18400}{35400}\right)^{\frac{1}{4}}$.

Step2: Calculate the value inside the parentheses

First, calculate $\frac{18400}{35400}\approx0.52$. Then find $\left(\frac{18400}{35400}\right)^{\frac{1}{4}}=\sqrt[4]{0.52}\approx0.84$.

Step3: Calculate the depreciation rate

$r = 1 - 0.84=0.16$. To convert to percentage, multiply by 100, so $r = 16.0%$.

Step4: Solve for V in terms of r

Start with $r = 1-\left(\frac{V}{35400}\right)^{\frac{1}{4}}$. Rearrange to get $\left(\frac{V}{35400}\right)^{\frac{1}{4}}=1 - r$. Raise both sides to the 4 - th power: $\frac{V}{35400}=(1 - r)^4$. Then $V = 35400(1 - r)^4$.

Answer:

a. $16.0%$ b. $V = 35400(1 - r)^4$