Question

an office machine is purchased for $4800. assume that its salvage value, v, in dollars, depreciates, according to a method called double - declining balance, by 40% each year and is given by v(t)=4800(0.60)^t where t is the time, in years, after purchase.
a) find v(6), and explain its meaning.
b) find v(6), and explain its meaning.
c) when will the salvage value of the office machine be half of the purchase price?
a) v(6) = $223.55
(round to the nearest cent as needed.)

Explanation:

Step1: Identify the depreciation formula

The salvage - value formula is $V(t)=4800(0.6)^{t}$, where $t$ is the time in years after purchase.

Step2: Find $V(6)$

Substitute $t = 6$ into the formula: $V(6)=4800\times(0.6)^{6}$.
$V(6)=4800\times0.046656 = 223.9488\approx223.95$. This means that 6 years after the purchase of the office machine, its salvage value is approximately $\$223.95$.

Step3: Find $V^{\prime}(t)$

First, recall the formula for the derivative of an exponential function $y = a\cdot b^{x}$, where $y^\prime=a\cdot b^{x}\ln(b)$. For $V(t)=4800(0.6)^{t}$, $V^{\prime}(t)=4800\times(0.6)^{t}\ln(0.6)$.
Substitute $t = 6$: $V^{\prime}(6)=4800\times(0.6)^{6}\ln(0.6)$.
$V^{\prime}(6)=223.9488\times(- 0.510826)\approx - 114.39$. This means that at 6 years after purchase, the salvage value of the office machine is decreasing at a rate of approximately $\$114.39$ per year.

Step4: Find when the salvage value is half of the purchase price

The purchase price is $\$4800$, and we want to find $t$ when $V(t)=2400$.
Set up the equation $4800(0.6)^{t}=2400$.
Divide both sides by 4800: $(0.6)^{t}=\frac{2400}{4800}=0.5$.
Take the natural - logarithm of both sides: $\ln((0.6)^{t})=\ln(0.5)$.
Using the property of logarithms $\ln(a^{b})=b\ln(a)$, we get $t\ln(0.6)=\ln(0.5)$.
Solve for $t$: $t=\frac{\ln(0.5)}{\ln(0.6)}\approx1.365$.

Answer:

a) $V(6)\approx223.95$. It means the salvage value of the office machine 6 years after purchase is about $\$223.95$.
b) $V^{\prime}(6)\approx - 114.39$. It means the salvage value is decreasing at a rate of about $\$114.39$ per year at 6 years after purchase.
c) $t\approx1.365$ years.