The minimum number of years needed for the vehicle's value to be less than $10,000 is 9 years
SOLUTION:
Given, Timothy paid $27,000 for a new truck.
The value of the truck depreciates 11% for each year of his ownership,
We have to find what is the minimum number of years needed for the vehicle's value to be less than $10,000.
Let the number of years be n.
Now we can use the below formula to find the minimum number of years.
[tex]\text { Present value }=\text { starting value } \times\left(1-\frac{\text {rate of depreciation}}{100}\right)^{\text {number of years. }}[/tex]
Now, present value < 10000
[tex]\begin{array}{l}{\text { Then, } 27000 \times\left(1-\frac{11}{100}\right)^{n}<10000} \\\\ {\left(1-\frac{11}{100}\right)^{n}<\frac{10000}{27000}} \\\\ {(1-0.11)^{n}<\frac{10}{27}} \\\\ {0.89^{n}<0.37037}\end{array}[/tex]
Now, let us use trail and error method, put n = 8
Then, 0.3936 < 0.3703, wrong, so put n = 9, ⇒ 0.3503 < 0.3703 correct.
Hence, it takes 9 years.
