Question

an office machine is purchased for $9100. assume that its salvage - value, v, in dollars, depreciates according to a method called double - declining balance, by 20% each year and is given by v(t)=9100(0.80)^t, where t is the time, in years, after purchase. a) find v(4) and explain its meaning. b) find v(4) and explain its meaning. c) when will the salvage value of the office machine be half of the purchase price?

Explanation:

Step1: Calculate V(4)

We are given the formula $V(t)=9100(0.80)^t$. Substitute $t = 4$ into the formula:
$V(4)=9100\times(0.80)^4=9100\times0.4096 = 3737.36$
This means that after 4 years, the salvage - value of the office machine is $3737.36$ dollars.

Step2: Calculate the derivative of V(t)

First, if $V(t)=9100(0.80)^t$, using the formula for the derivative of an exponential function $y = a\cdot b^x$ where $y^\prime=a\cdot b^x\ln(b)$. Here $a = 9100$ and $b = 0.80$. So $V^\prime(t)=9100\times(0.80)^t\ln(0.80)$.
Substitute $t = 4$ into $V^\prime(t)$:
$V^\prime(4)=9100\times(0.80)^4\times\ln(0.80)=3737.36\times(- 0.2231)=-834.74$
The expression $V^\prime(4)$ means that after 4 years, the salvage - value of the office machine is decreasing at a rate of $834.74$ dollars per year.

Step3: Find when the salvage value is half of the purchase price

The purchase price is $9100$ dollars. We want to find $t$ when $V(t)=\frac{9100}{2}=4550$.
Set $V(t)=4550$, so $4550 = 9100(0.80)^t$.
Divide both sides by $9100$: $\frac{4550}{9100}=(0.80)^t$, which simplifies to $0.5=(0.80)^t$.
Take the natural logarithm of both sides: $\ln(0.5)=t\ln(0.80)$.
Solve for $t$: $t=\frac{\ln(0.5)}{\ln(0.80)}=\frac{-0.6931}{-0.2231}\approx3.11$ years.

Answer:

a) $V(4)=3737.36$. After 4 years, the salvage - value of the office machine is $3737.36$ dollars.
b) $V^\prime(4)=-834.74$. After 4 years, the salvage - value of the office machine is decreasing at a rate of $834.74$ dollars per year.
c) Approximately $3.11$ years.