Question

two months ago, ava bought a tablet for $900. the warranty note says the tablet depreciates by 25% every 6 months. ava plans to trade it in after 2 years. what will be the value v of the tablet after 2 years? a. $v = 900(0.75)^{2}$ $v = $506.25$ b. $v = 900(0.75)^{4}$ $v = $284.77$ c. $v = 900(0.25)^{4}$ $v = $3.52$ d. $v = 900(0.75)^{8}$ $v = $90.10$

Explanation:

Step1: Determine the number of depreciation periods

The tablet depreciates every 6 months, and we need to find the value after 2 years. Since 1 year has 12 months, 2 years have \( 2\times12 = 24 \) months. The number of 6 - month periods in 24 months is \( \frac{24}{6}=4 \) periods.

Step2: Determine the depreciation factor

If the tablet depreciates by 25% every 6 months, the remaining value after each 6 - month period is \( 1 - 0.25=0.75 \) of the previous value.

Step3: Use the exponential depreciation formula

The formula for exponential depreciation is \( V = P(1 - r)^n \), where \( P \) is the initial value, \( r \) is the rate of depreciation per period, and \( n \) is the number of periods. Here, \( P = 900 \), \( r = 0.25 \), and \( n = 4 \). So the formula becomes \( V=900(0.75)^4 \).

Step4: Calculate the value

First, calculate \( 0.75^4=(0.75\times0.75)\times(0.75\times0.75)=0.5625\times0.5625 = 0.31640625 \)

Then, \( V = 900\times0.31640625=284.765625\approx284.77 \)

Answer:

B. \( V = 900(0.75)^4 \), \( V=\$284.77 \)