Question

the amount of credit - card spending during the holiday period has increased by 11% per year since 1987. the amount, a, in billions of dollars of credit - card spending during the holiday period in a given year can be modeled by ( a = f(x)=36.3e^{0.11x} ), where x represents the number of years since 1987. what is the doubling time? the doubling time is approximately ( square ) years. (round to two decimal places as needed.)

Explanation:

Step1: Set up the equation for doubling

We want to find \(x\) when \(A = 2\times36.3\). Substitute into the formula \(A = 36.3e^{0.11x}\), so \(2\times36.3=36.3e^{0.11x}\).
Divide both sides by \(36.3\): \(2 = e^{0.11x}\).

Step2: Take the natural logarithm of both sides

Using the property \(\ln(e^{a})=a\), if \(2 = e^{0.11x}\), then \(\ln(2)=\ln(e^{0.11x})\).
So \(\ln(2)=0.11x\).

Step3: Solve for \(x\)

We know that \(\ln(2)\approx0.6931\). Then \(x=\frac{\ln(2)}{0.11}\).
Substitute \(\ln(2)\approx0.6931\) into the formula: \(x=\frac{0.6931}{0.11}\approx6.30\).

Answer:

\(6.30\)