21. If a pane of glass absorbs $$ 10 \% $$ of the light passing through it, then the percent $$ P $$ of light that passes through $$ n $$ successive panes is given by the approximation $$ P(n)=100 e^{-1 / n} $$. How many panes of glass are required to block at least $$ 60 \% $$ of the light?

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To find the number of panes of glass required to block at least $$ 60\% $$ of the light, we first need to determine the percentage of light that passes through the panes. If $$ P(n) $$ is the percentage of light that passes through $$ n $$ panes, then blocking at least $$ 60\% $$ of the light means that at most $$ 40\% $$ of the light should pass through.

We start with the equation given:

$$
P(n) = 100 e^{-1/n}
$$

We want to find $$ n $$ such that:

$$
P(n) \leq 40
$$

This translates to:

$$
100 e^{-1/n} \leq 40
$$

Dividing both sides by $$ 100 $$:

$$
e^{-1/n} \leq 0.4
$$

Next, we take the natural logarithm of both sides:

$$
- \frac{1}{n} \leq \ln(0.4)
$$

Since $$ \ln(0.4) $$ is negative, we can multiply both sides by $$ -1 $$ (which reverses the inequality):

$$
\frac{1}{n} \geq -\ln(0.4)
$$

Taking the reciprocal gives:

$$
n \leq \frac{1}{-\ln(0.4)}
$$

Now we calculate $$ -\ln(0.4) $$:

$$
\ln(0.4) \approx -0.91629 \quad \text{(using a calculator)}
$$

Thus,

$$
-\ln(0.4) \approx 0.91629
$$

Now we can find $$ n $$:

$$
n \leq \frac{1}{0.91629} \approx 1.0905
$$

Since $$ n $$ must be a whole number, we round up to the nearest whole number, which is $$ 2 $$.

To confirm, we can check $$ P(2) $$:

$$
P(2) = 100 e^{-1/2} \approx 100 \cdot 0.60653 \approx 60.653
$$

This means that with $$ 2 $$ panes, approximately $$ 60.653\% $$ of the light passes through, which is slightly more than $$ 60\% $$.

Now, checking $$ P(3) $$:

$$
P(3) = 100 e^{-1/3} \approx 100 \cdot 0.71653 \approx 71.653
$$

This means that with $$ 3 $$ panes, approximately $$ 71.653\% $$ of the light passes through, which is more than $$ 60\% $$.

Thus, the minimum number of panes required to block at least $$ 60\% $$ of the light is:

$$
\boxed{2}
$$
The minimum number of panes required to block at least $$ 60\% $$ of the light is 2.

If a pane of glass absorbs of the light passing through it, then the
percent of light that passes through successive panes is given by the
approximation . How many panes of glass are required to block at
least of the light?

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If a pane of glass absorbs of the light passing through it, then the percent of light that passes through successive panes is given by the approximation . How many panes of glass are required to block at least of the light?