A computer purchased for \( \$ 550 \) loses \( 10 \% \) of its value every year. The computer's value can be modeled by the function \( v(t)=a \cdot b^{t} \), where \( v \) is the dollar value and \( t \) the number of years since purchase. (A) In the exponential model \( a=\square \) and \( b=\square \) (B) In how many years will the computer be worth half its original value? Round answer to 1 decimal place. The answer is Question Help: Video

To solve the problem, we will follow the steps outlined: ### Step 1: Extract Known Conditions - The initial value of the computer is \( \$550 \). - The computer loses \( 10\% \) of its value every year. ### Step 2: Determine the Exponential Model Parameters The value of the computer can be modeled by the function \( v(t) = a \cdot b^{t} \). #### (A) Identify \( a \) and \( b \) - The initial value \( a \) is the value of the computer at \( t = 0 \), which is \( 550 \). - The value decreases by \( 10\% \) each year, meaning it retains \( 90\% \) of its value. Therefore, \( b = 1 - 0.10 = 0.90 \). Thus, we have: - \( a = 550 \) - \( b = 0.90 \) ### Step 3: Find the Time When the Value is Half We need to find \( t \) when the value \( v(t) \) is half of the original value: \[ v(t) = \frac{550}{2} = 275 \] Substituting into the model: \[ 275 = 550 \cdot (0.90)^{t} \] Now, we will solve for \( t \): 1. Divide both sides by \( 550 \): \[ \frac{275}{550} = (0.90)^{t} \] \[ 0.5 = (0.90)^{t} \] 2. Take the logarithm of both sides: \[ \log(0.5) = t \cdot \log(0.90) \] 3. Solve for \( t \): \[ t = \frac{\log(0.5)}{\log(0.90)} \] Now, let's calculate \( t \). Calculate the value by following steps: - step0: Calculate: \(\frac{\log_{10}{\left(0.5\right)}}{\log_{10}{\left(0.9\right)}}\) - step1: Calculate: \(\frac{-\log_{10}{\left(2\right)}}{\log_{10}{\left(0.9\right)}}\) - step2: Convert the expressions: \(\frac{-\log_{10}{\left(2\right)}}{\log_{10}{\left(\frac{9}{10}\right)}}\) - step3: Use the logarithm base change rule: \(-\log_{\frac{9}{10}}{\left(2\right)}\) The calculation gives us: \[ t \approx 6.578813 \] Rounding this to one decimal place, we find: \[ t \approx 6.6 \] ### Final Answers (A) In the exponential model: - \( a = 550 \) - \( b = 0.90 \) (B) The computer will be worth half its original value in approximately \( 6.6 \) years.