Question

attempt 1: 10 attempts remaining. the resale value v(t) of a car, in dollars, t years after purchase is modeled by: v(t)=\sqrt{-2900t + 38000} find the rate at which the cars value is changing after 7 years. (round your answer to the nearest cent.) v(7)=

Explanation:

Step1: Rewrite the function

Rewrite $V(t)=\sqrt{- 2900t + 38000}=(-2900t + 38000)^{\frac{1}{2}}$.

Step2: Apply the chain - rule

The chain - rule states that if $y = u^{\frac{1}{2}}$ and $u=-2900t + 38000$, then $\frac{dy}{dt}=\frac{dy}{du}\cdot\frac{du}{dt}$. First, find $\frac{dy}{du}$: $\frac{d}{du}(u^{\frac{1}{2}})=\frac{1}{2}u^{-\frac{1}{2}}$. Second, find $\frac{du}{dt}$: $\frac{d}{dt}(-2900t + 38000)=-2900$. Then $V'(t)=\frac{1}{2}(-2900t + 38000)^{-\frac{1}{2}}\times(-2900)=\frac{-2900}{2\sqrt{-2900t + 38000}}=\frac{-1450}{\sqrt{-2900t + 38000}}$.

Step3: Evaluate at $t = 7$

Substitute $t = 7$ into $V'(t)$: $V'(7)=\frac{-1450}{\sqrt{-2900\times7 + 38000}}=\frac{-1450}{\sqrt{-20300+38000}}=\frac{-1450}{\sqrt{17700}}\approx\frac{-1450}{133.0413}\approx - 10.90$.

Answer:

$-10.90$