Let's go through the problem step-by-step:
### Part (A)
To write the exponential model for the value of the computer over time, we need to determine the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex].
1. Initial Value ([tex]\( a \)[/tex]):
- The initial value of the computer when purchased is [tex]$750. Therefore, \( a = 750 \).
2. Decay Factor (\( b \)):
- The computer loses \( 17\% \) of its value each year. This means it retains \( 83\% \) of its value each year (since \( 100\% - 17\% = 83\%\)).
- To express \( 83\% \) as a decimal for our function, we write \( 83\% \) as \( 0.83 \).
- Therefore, the decay factor \( b = 0.83 \).
So, the exponential model can be written as: \( v(t) = 750 \cdot 0.83^t \).
### Part (B)
Next, we need to determine how many years it will take for the computer to be worth half its original value. The original value is $[/tex]750, so half of this value is $375.
We set up the equation:
[tex]\[
375 = 750 \cdot 0.83^t
\][/tex]
To solve for [tex]\( t \)[/tex]:
1. Divide both sides by 750:
[tex]\[
\frac{375}{750} = 0.83^t
\][/tex]
[tex]\[
0.5 = 0.83^t
\][/tex]
2. Take the natural logarithm of both sides to solve for [tex]\( t \)[/tex]:
[tex]\[
\ln(0.5) = \ln(0.83^t)
\][/tex]
3. Use the logarithm power rule [tex]\(\ln(a^b) = b \cdot \ln(a)\)[/tex]:
[tex]\[
\ln(0.5) = t \cdot \ln(0.83)
\][/tex]
4. Solve for [tex]\( t \)[/tex]:
[tex]\[
t = \frac{\ln(0.5)}{\ln(0.83)}
\][/tex]
5. Simplify the expression:
After calculating the above expression, we find that:
[tex]\[ t \approx 3.7 \][/tex]
### Final Answer
(A) For the exponential model, [tex]\( a = 750 \)[/tex] and [tex]\( b = 0.83 \)[/tex].
(B) The computer will be worth half its original value in approximately [tex]\( 3.7 \)[/tex] years.
