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 left parenthesis n right parenthesis equals 100 left parenthesis 0.97 right parenthesis Superscript n
​(a) What percent of light will pass through 5 ​panes?
​(b) What percent of light will pass through 10 ​panes?
​(c) Explain the meaning of the base 0.97 in this problem.
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Part 1
​(a) The percent of light that will pass through 5 panes is approximately

enter your response here​%.
​(Round to the nearest whole number as​ needed.)

We are given the function for the percentage of light that passes through \( n \) successive panes of glass:

\[
p(n) = 100 \times (0.97)^n
\]

### (a) The percent of light that will pass through 5 panes:
Substitute \( n = 5 \) into the equation:

\[
p(5) = 100 \times (0.97)^5
\]

First, calculate \( (0.97)^5 \):

\[
(0.97)^5 \approx 0.8587
\]

Now multiply by 100 to get the percentage:

\[
p(5) = 100 \times 0.8587 = 85.87
\]

Rounding to the nearest whole number:

\[
p(5) \approx 86\%
\]

So, approximately **86%** of light will pass through 5 panes.

---

### (b) The percent of light that will pass through 10 panes:
Substitute \( n = 10 \) into the equation:

\[
p(10) = 100 \times (0.97)^{10}
\]

First, calculate \( (0.97)^{10} \):

\[
(0.97)^{10} \approx 0.7374
\]

Now multiply by 100:

\[
p(10) = 100 \times 0.7374 = 73.74
\]

Rounding to the nearest whole number:

\[
p(10) \approx 74\%
\]

So, approximately **74%** of light will pass through 10 panes.

---

### (c) The meaning of the base 0.97:
The base \( 0.97 \) represents the percentage of light that passes through a **single** pane of glass. Specifically, it means that 97% of the light passes through one pane, and 3% of the light is absorbed or blocked by the pane. The exponential function then models how this 97% transmission factor compounds as light passes through multiple panes.

---

Do you have any further questions or want more details?

Here are some related questions to expand on this:

1. How does the percent of light change as the number of panes increases?
2. What would happen to the light transmission if each pane blocked 5% of the light?
3. Can you derive the percent of light that passes through 15 panes?
4. How would the function change if 4% of light was absorbed per pane?
5. What is the total percentage of light blocked by 8 panes?

**Tip:** In problems involving successive multiplications (like repeated percentages), exponential functions are often used to model the compound effect!

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 10 panes?
(c) Explain the meaning of the base 0.97 in this problem.

Calculate the percentage of light that passes through 5 and 10 successive panes of glass, each blocking 3% of light. Learn about exponential decay and how to interpret the base of a function in this problem suitable for Grades 9-10.

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