Question

ova v. math of finance sem a- a
tomobile ownership
a high school music department buys a piano for $3, 000. it is expected to have a salvage value of $500 after 10 years.
what equation represents its straight-line depreciation?
$d(t) = 3,000 - 25t$
$d(t) = 3,000 + 25t$
$d(t) = 3,000 - 30t$
$d(t) = 2,500 - 25t$

Explanation:

Step1: Recall straight - line depreciation formula

The straight - line depreciation formula is \( D(t)=V_0 - rt \), where \( V_0 \) is the initial value, \( r \) is the rate of depreciation per unit time, and \( t \) is the time.
The initial value of the piano \( V_0=\$3000 \), and after \( t = 10 \) years, the salvage value \( D(10)=\$500 \).

Step2: Calculate the depreciation rate \( r \)

We know that \( D(10)=3000-10r = 500 \).
Subtract 3000 from both sides: \( - 10r=500 - 3000=-2500 \).
Divide both sides by - 10: \( r=\frac{- 2500}{-10}=250 \)? Wait, no, wait. Wait, the salvage value is 500 after 10 years. The total depreciation over 10 years is \( 3000 - 500=2500 \). So the annual depreciation \( r=\frac{2500}{10} = 250 \)? Wait, no, the options have 25 or 30. Wait, maybe I misread. Wait, the options are \( D(t)=3000 - 25t \), \( D(t)=3000 + 25t \), \( D(t)=3000-30t \), \( D(t)=2500 - 25t \). Wait, let's recalculate.
The depreciation per year: The value decreases from 3000 to 500 in 10 years. The change in value is \( 3000 - 500 = 2500 \) over 10 years. So the annual depreciation is \( \frac{2500}{10}=250 \)? But the options have 25. Wait, maybe the problem has a typo, but among the given options, the correct form of straight - line depreciation (decreasing function, so subtraction) with initial value 3000 is \( D(t)=3000 - 25t \) (assuming the rate is 25, maybe the salvage value is miscalculated in the problem, but from the options, the only one with initial value 3000 and a decreasing function is \( D(t)=3000 - 25t \) (since \( D(t)=3000 + 25t \) is increasing, which is not depreciation, \( D(t)=3000-30t \) would give a different salvage value, and \( D(t)=2500 - 25t \) has initial value 2500 which is wrong as initial value is 3000). So the correct equation is \( D(t)=3000 - 25t \).

Answer:

\( D(t) = 3,000 - 25t \)