A computer purchased for \( \$ 1,200 \) loses \( 12 \% \) 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 \( b= \) (B) In how many years will the computer be worth half its original value? Round answer to 1 decimal place. The answer is Question 7

To set up the exponential model, we have the initial value \( a = 1200 \) since that’s the price paid for the computer. The value decreases by \( 12\% \) each year, meaning it retains \( 88\% \) of its value, which can be represented as the decimal \( b = 0.88 \). Now, don't be discouraged; let's tackle when the computer will be worth half its original value. Half of the original \( \$1200 \) is \( \$600 \). We can set the equation \( v(t) = 1200 \cdot 0.88^t \), and we want to find \( t \) such that \( 1200 \cdot 0.88^t = 600 \). Solving for \( t \) gives: \[ 0.88^t = \frac{600}{1200} = 0.5 \] Taking the logarithm of both sides: \[ t \cdot \log(0.88) = \log(0.5) \] Calculating \( t \): \[ t = \frac{\log(0.5)}{\log(0.88)} \approx \frac{-0.3010}{-0.0530} \approx 5.7 \] So, it will take approximately \( 5.7 \) years for the computer to lose half its value! Now, for some fun tidbits! Did you know that the depreciation of electronics can often be much steeper than that of other types of assets? In fact, a new smartphone can lose up to \( 30\% \) of its value as soon as you unbox it! So if you're in the market for tech, timing your purchase wisely can save you a bundle! Additionally, there's a fascinating world behind how companies decide product prices and their depreciation rates! Many factors play into it, including market demand, material quality, and technological advances. If you're curious about the economics of technology, books on behavioral economics provide insight into consumer choices and price elasticity which reflect how we value gadgets over time!