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

The initial value \( a \) of the computer is \( 550 \) (the purchase price), so \( a = 550 \). The value decreases by \( 10\% \) each year, meaning it retains \( 90\% \) of its value after a year; hence, \( b = 0.9 \). This gives us the model \( v(t) = 550 \cdot 0.9^t \). To determine when the computer will be worth half its original value, set \( v(t) = 275 \) (which is half of \( 550 \)): \[ 275 = 550 \cdot 0.9^t \] Dividing both sides by \( 550 \) gives: \[ 0.5 = 0.9^t \] Taking the logarithm of both sides: \[ \log(0.5) = t \cdot \log(0.9) \] Now solving for \( t \): \[ t \approx \frac{\log(0.5)}{\log(0.9)} \approx 6.58 \] Rounding to one decimal place, it will take approximately \( 6.6 \) years for the computer to be worth half its original value. --- Did you know that the concept of exponential decay isn't just limited to computers? It's also applicable in natural sciences, such as population decline in certain species or radioactive decay of elements! Just like any investment, staying informed about the factors that influence value depreciation can aid in making better purchasing decisions. Exponential decay is such a savvy tool for managing finances! To make it even more relatable, think about how consumer electronics seem to lose value faster than you can say "new model!" Keep in mind that proper care, frequent upgrades, and using the latest software can prolong the life and usability of your device, maintaining more of its value over time!