A computer purchased for $\$ 800$ loses $10 \%$ of its value every year. The computer's value can be modeled by therfunction $v(t)=a \cdot b^{t}$, where $v$ is the dollar value and $t$ the number of years since purchase.
(A) Give the function that models the decrease in value of the computer: $v(t)=$
(B) In how many years will the computer be worth half its original value? Round answer to 1 decimal place.
Answer
Final Answer: \n(A) The function that models the decrease in value of the computer is \(v(t) = 800 \cdot 0.9^{t}\).\n(B) The computer will be worth half its original value after approximately \(\boxed{6.6}\) years.
Step 1 ：The problem is asking for two things. First, it wants the function that models the decrease in value of the computer. This is a simple application of the formula given in the problem. The initial value of the computer, \(a\), is \$800. The rate of decrease, \(b\), is \(90\%\) or \(0.9\) (since it loses \(10\%\) of its value every year). So the function is \(v(t) = 800 \cdot 0.9^{t}\).
Step 2 ：Second, it wants to know when the computer will be worth half its original value. This is a matter of solving the equation \(800 \cdot 0.9^{t} = 400\) for \(t\).
Step 3 ：The calculation gives that the computer will be worth half its original value after approximately 6.578813478960585 years.
Step 4 ：However, the question asks for the answer to be rounded to 1 decimal place. So, the rounded value is 6.6 years.
Step 5 ：Final Answer: \n(A) The function that models the decrease in value of the computer is \(v(t) = 800 \cdot 0.9^{t}\).\n(B) The computer will be worth half its original value after approximately \(\boxed{6.6}\) years.
