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.

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(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.

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(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.

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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!