Question

a laptop computer is purchased for $2000. each year, its value is 75% of its value the year before. after how many years will the laptop computer be worth $500 or less? (use the calculator provided if necessary.) write the smallest possible whole number answer.

Explanation:

Step1: Define the depreciation formula

The value of the laptop after \( n \) years follows the formula \( V(n)=2000\times(0.75)^n \), where \( V(n) \) is the value after \( n \) years. We need to find the smallest integer \( n \) such that \( V(n)\leq500 \).

Step2: Set up the inequality

We set up the inequality \( 2000\times(0.75)^n\leq500 \). Divide both sides by 2000: \( (0.75)^n\leq\frac{500}{2000}=\frac{1}{4} = 0.25 \).

Step3: Solve for \( n \) using logarithms

Take the natural logarithm of both sides: \( \ln((0.75)^n)\leq\ln(0.25) \). Using the logarithm power rule \( \ln(a^b)=b\ln(a) \), we get \( n\ln(0.75)\leq\ln(0.25) \). Since \( \ln(0.75) \) is negative (because \( 0.75<1 \)), when we divide both sides by \( \ln(0.75) \), the inequality sign flips: \( n\geq\frac{\ln(0.25)}{\ln(0.75)} \).

Calculate \( \frac{\ln(0.25)}{\ln(0.75)}\approx\frac{-1.3863}{-0.2877}\approx4.82 \).

Since \( n \) must be a whole number and \( n\geq4.82 \), the smallest whole number \( n \) is 5.

Answer:

5