Question

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 $$ p(n)=100 \cdot 0.97^{n} $$ (a) What percent of light will pass through 10 panes? (b) What percent of light will pass through 25 panes? (c) Explain the meaning of the base 0.97 in this problem.

Short answer

(a) 73.74% (b) 47.67% (c) Each pane allows 97% of the light to pass through.

Step-by-step solution

- Identify the formula

Use the given formula for the percentage of light passing through multiple panes of glass: \[ p(n) = 100 \times 0.97^n \]

- Calculate for 10 panes

Plug in \( n = 10 \) into the formula to find the percent of light passing through 10 panes.\[ p(10) = 100 \times 0.97^{10} \]Using a calculator: \[ 0.97^{10} \approx 0.7374 \]Thus, \[ p(10) = 100 \times 0.7374 = 73.74\text{ percent} \]

- Calculate for 25 panes

Plug in \( n = 25 \) into the formula to find the percent of light passing through 25 panes.\[ p(25) = 100 \times 0.97^{25} \]Using a calculator: \[ 0.97^{25} \approx 0.4767 \]Thus, \[ p(25) = 100 \times 0.4767 = 47.67\text{ percent} \]

- Explain the base 0.97

The base 0.97 in the formula represents the proportion of light that passes through a single pane of glass. Since each pane allows 97% of the light to pass through, the multiplicative factor is 0.97 for each additional pane.

Key concepts

percentage of light

When we talk about the 'percentage of light' passing through panes of glass, we're essentially discussing how much light is left after some gets blocked. Each pane of glass stops a small portion of the light. Specifically, in this problem, each pane blocks 3% of the light, meaning 97% of the light gets through.
The percentage of light passing through multiple panes can be calculated using the formula:
\( p(n) = 100 \times 0.97^n \).
This formula helps us quickly find out how much light will pass through as we add more panes. The more panes you have, the more light gets absorbed and the less light makes it through.

successive panes

Successive panes mean one pane after another in sequence. Think of it like stacking multiple window glass panes and seeing how much light can go through all of them together.
In our formula \( p(n) = 100 \times 0.97^n \), 'n' represents the number of successive panes.
Each pane lets only 97% of the light through, so with each additional pane, this 97% reduction happens again and again.
For example, after the first pane, 97% of the light is left. After the second pane, 97% of that remaining light gets through, and so on.
This repeated reduction is why we have an exponential function to describe the overall percentage of light that makes it through multiple panes.

exponential function

An exponential function shows how a quantity changes at a constant multiplicative rate over time.
Our formula, \( p(n) = 100 \times 0.97^n \), is an exponential function because the percentage of light decreases by a factor of 0.97 for each additional pane.
The key feature of an exponential function is that the rate of change is proportional to the current quantity. Here, after each pane, 97% of the remaining light passes through, so the light decreases exponentially.
This makes exponential functions very powerful in describing natural processes where a quantity reduces (or grows) step-by-step, such as radioactive decay, population growth, and our light transmission problem.

base explanation

In our formula \( p(n) = 100 \times 0.97^n \), the base of the exponent is 0.97.
This base tells us the rate at which light passes through each individual pane. Specifically, 0.97 means that each pane lets 97% of the light through, blocking the remaining 3%.
As you add more panes, this base is raised to a higher exponent (number of panes), which continuously reduces the total percentage of light.
The base must always be a number between 0 and 1 in exponential decay problems to indicate a reduction. If it were greater than 1, we'd see an exponential increase instead.
Understanding the base helps us grasp how quickly the light diminishes as it passes through successive panes.