Answer:
[tex] 27.75\% [/tex]
Step-by-step explanation:
To find the total percentage depreciation of the boat's value after two years, we can use the formula for compound interest (or depreciation, in this case):
[tex] A = P(1 - r)^n [/tex]
Where:
[tex] A [/tex] is the final amount (or value in this case),
[tex] P [/tex] is the initial amount (or value in this case),
[tex] r [/tex] is the rate of depreciation (as a decimal),
[tex] n [/tex] is the number of years.
Given that the boat depreciates by 15% each year, [tex] r = 0.15 [/tex] (as a decimal).
Let's denote the initial value of the boat as [tex] P [/tex].
For the first year:
[tex] A_1 = P(1 - 0.15) = P(0.85) [/tex]
For the second year:
[tex] A_2 = (0.85P)(1 - 0.15) = (0.85P)(0.85) = 0.85^2P [/tex]
The total percentage depreciation after two years can be calculated by finding the ratio of the final value to the initial value and then expressing it as a percentage:
[tex] \textsf{Total Depreciation Percentage} = \left(1 - \dfrac{A_2}{P}\right) \times 100\% [/tex]
Let's calculate it:
[tex] \textsf{Total Depreciation Percentage} = \left(1 - \dfrac{0.85^2P}{P}\right) \times 100\% [/tex]
[tex] = (1 - 0.85^2) \times 100\% [/tex]
[tex] \approx (1 - 0.7225) \times 100\% [/tex]
[tex] \approx 0.2775 \times 100\% [/tex]
[tex] \approx 27.75\% [/tex]
So, the total percentage by which the value of the boat had depreciated by the end of the second year after Pam bought the boat is approximately [tex] 27.75\% [/tex].
