To find the percentage of pine trees in the forest that are taller than 100 feet, we need to follow these steps:
1. Determine the z-score: The z-score is a measure of how many standard deviations a value is from the mean. It is calculated using the formula:
[tex]\[
z = \frac{{X - \mu}}{{\sigma}}
\][/tex]
where:
- [tex]\(X\)[/tex] is the value of interest (in this case, 100 feet),
- [tex]\(\mu\)[/tex] is the mean (86 feet),
- [tex]\(\sigma\)[/tex] is the standard deviation (8 feet).
Plugging in the values:
[tex]\[
z = \frac{{100 - 86}}{{8}} = \frac{14}{8} = 1.75
\][/tex]
2. Find the cumulative probability: The next step is to use the z-score to find the cumulative probability from the z-table. The z-score value we have is 1.75.
From the provided z-table, we find the corresponding cumulative probability:
[tex]\[
P(Z < 1.75) \approx 0.95994
\][/tex]
This means that approximately 95.994% of the trees are shorter than 100 feet.
3. Calculate the percentage of trees taller than 100 feet: The percentage of trees taller than 100 feet is the complement of the cumulative probability. This can be calculated as:
[tex]\[
P(X > 100) = 1 - P(Z < 1.75)
\][/tex]
So:
[tex]\[
P(X > 100) = 1 - 0.95994 = 0.04006
\][/tex]
4. Convert the result to a percentage: To express the result as a percentage, multiply by 100:
[tex]\[
0.04006 \times 100 \approx 4.006\%
\][/tex]
Therefore, approximately [tex]\(4\%\)[/tex] of the pine trees in the forest are taller than 100 feet.
So, the correct answer is:
[tex]\[
\boxed{4 \%}
\][/tex]
