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(0.8)^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?

1. v(4) = $5727.30

(round to the nearest cent as needed.)
the expression v(4) means that after 4 year(s), the salvage value is v(4) dollars.
b) v(4) = $
(round to the nearest cent as needed.)

Explanation:

Step1: Calculate V(4)

We are given the formula $V(t)=9100(0.8)^t$. Substitute $t = 4$ into the formula:
$V(4)=9100\times(0.8)^4=9100\times0.4096 = 3727.36$

Step2: Interpret V(4)

The expression $V(4)$ represents the salvage - value of the office machine 4 years after its purchase. It is $3727.36$ dollars.

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 up the equation $4550 = 9100(0.8)^t$.
First, divide both sides of the equation by $9100$:
$\frac{4550}{9100}=(0.8)^t$, so $0.5=(0.8)^t$.
Take the natural logarithm of both sides: $\ln(0.5)=\ln((0.8)^t)$.
Using the property of logarithms $\ln(a^b)=b\ln(a)$, we get $\ln(0.5)=t\ln(0.8)$.
Solve for $t$: $t=\frac{\ln(0.5)}{\ln(0.8)}\approx3.1063$.

Answer:

a) $V(4) = 3727.36$. The meaning of $V(4)$ is that the salvage value of the office machine 4 years after purchase is $3727.36$ dollars.
b) It seems there is a mis - label in part b as it is the same as part a. If it was meant to be something else, please clarify. But based on the current problem, the answer is the same as part a.
c) Approximately $3.11$ years (rounded to two decimal places).