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?
a) v(4) = $3727.36
(round to the nearest cent as needed.)
the expression v(4) means that after year(s), the salvage value

Explanation:

Step1: Identify the depreciation formula

The formula for the salvage - value is $V(t)=9100(0.8)^t$, where $t$ is the time in years after purchase.

Step2: Calculate $V(4)$

Substitute $t = 4$ into the formula: $V(4)=9100\times(0.8)^4$. First, calculate $(0.8)^4=0.8\times0.8\times0.8\times0.8 = 0.4096$. Then, $V(4)=9100\times0.4096 = 3727.36$.

Step3: Interpret $V(4)$

The expression $V(4)$ means that after 4 years, the salvage value of the office machine is $\$3727.36$.

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

The purchase price is $\$9100$, and we want to find $t$ when $V(t)=\frac{9100}{2}=4550$. So we set up the equation $4550 = 9100(0.8)^t$. Divide both sides of the equation by $9100$: $\frac{4550}{9100}=(0.8)^t$, which simplifies to $0.5=(0.8)^t$. 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 $\ln(0.5)=t\ln(0.8)$. Then $t=\frac{\ln(0.5)}{\ln(0.8)}=\frac{- 0.693147}{-0.223144}\approx3.11$.

Answer:

a) $V(4) = 3727.36$. The expression $V(4)$ means that after 4 years, the salvage value of the office machine is $\$3727.36$.
c) The salvage value of the office machine will be half of the purchase price after approximately $3.11$ years.