Question

an office machine is purchased for $7700. 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)=7700(0.6)^t, where t is the time, in years, after purchase.
a) find v(3) and explain its meaning.
b) find v(3) and explain its meaning.
c) when will the salvage value of the office machine be half of the purchase price?
a) v(3) = $1003.20 (round to the nearest cent as needed.) the expression v(3) means that after year(s), the salvage value

Explanation:

Step1: Calculate V(3)

The formula for the salvage - value is $V(t)=7700(0.6)^t$. Substitute $t = 3$ into the formula:
$V(3)=7700\times(0.6)^3=7700\times0.216 = 1663.20$
The expression $V(3)$ means that after 3 years, the salvage value of the office machine is $1663.20$ dollars.

Step2: Calculate when the salvage - value is half of the purchase price

The purchase price is $7700$ dollars. Half of the purchase price is $\frac{7700}{2}=3850$ dollars.
Set $V(t)=3850$ in the formula $V(t)=7700(0.6)^t$.
We get the equation $3850 = 7700(0.6)^t$.
First, divide both sides of the equation by $7700$:
$\frac{3850}{7700}=(0.6)^t$
$0.5=(0.6)^t$.
Take the natural logarithm of both sides: $\ln(0.5)=t\ln(0.6)$.
Then solve for $t$: $t=\frac{\ln(0.5)}{\ln(0.6)}\approx1.3652$ years.

Answer:

a) $V(3) = 1663.20$. The expression $V(3)$ means that after 3 years, the salvage value of the office machine is $1663.20$ dollars.
c) $t\approx1.37$ years (rounded to two decimal places).