Problem
If a single pane of glass obliterates $3 \%$ of the light passing through it, the percent $p$ of light that passes through $n$ successive panes is given approximately by the function below. \[ p(n)=100(0.97)^{n} \] (a) What percent of light will pass through 5 panes? (b) What percent of light will pass through 30 panes? (c) Explain the meaning of the base 0.97 in this problem. (a) The percent of light that will pass through 5 panes is approximately $\%$. (Round to the nearest whole number as needed.)
Solution
Step 1 ：Substitute \(n = 5\) into the function \(p(n)\): \(p(5) = 100(0.97)^5\)
Step 2 ：Calculate the value: \(p(5) = 100 \times 0.8587340257 \approx 85.87\)
Step 3 ：\(\boxed{86}\)% (rounded to the nearest whole number) of the light will pass through 5 panes
