Question

an office machine is purchased for $4100. 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)=4100(0.60)^t, where t is the time in years, after purchase.
a) find v(1) 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(1) = $2460 (round to the nearest cent as needed.) the expression v(1) means that after 1 year(s), the salvage value is v(1) dollars.
b) v(6) = - 114.60 (round to the nearest cent as needed.) the expression v(6) means that after 6 year(s), the salvage value is changing by v(6) dollars per year.
c) the salvage value of the office machine will be half of the purchase price after years. (round to two decimal places as needed.)

Explanation:

Step1: Recall the depreciation formula

The formula for the salvage - value $V(t)$ of an item depreciating by a double - declining balance method is $V(t)=4000(0.6)^t$, where the initial value is $4000$ and the depreciation rate is $40\%$ (so the remaining value factor each year is $1 - 0.4=0.6$).

Step2: Calculate $V(1)$

Substitute $t = 1$ into the formula $V(t)=4000(0.6)^t$. So $V(1)=4000\times0.6^1=2400$. The meaning of $V(1)$ is the salvage value of the office machine 1 year after purchase.

Step3: Calculate $V(6)$

Substitute $t = 6$ into the formula $V(t)=4000(0.6)^t$. So $V(6)=4000\times(0.6)^6=4000\times0.046656 = 186.624\approx186.62$. The meaning of $V(6)$ is the salvage value of the office machine 6 years after purchase.

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

The purchase price is $4000$, and we want to find $t$ when $V(t)=\frac{4000}{2}=2000$. Set up the equation $2000 = 4000(0.6)^t$. First, divide both sides of the equation by $4000$ to get $\frac{2000}{4000}=(0.6)^t$, which simplifies to $0.5=(0.6)^t$. Then, take the natural logarithm of both sides: $\ln(0.5)=\ln((0.6)^t)$. Using the property of logarithms $\ln(a^b)=b\ln(a)$, we have $\ln(0.5)=t\ln(0.6)$. Solve for $t$: $t=\frac{\ln(0.5)}{\ln(0.6)}\approx1.36$.

Answer:

a) $V(1) = 2400$. The meaning is the salvage value of the office machine 1 year after purchase.
b) $V(6)\approx186.62$. The meaning is the salvage value of the office machine 6 years after purchase.
c) $t\approx1.36$ years.