Question
A 24 foot tall tree casts a shadow that is 10 feet long. What is the distance to the top of its shadow? Apply a Pythagorean triple to solve this problem
Answers
GPT-4o mini
To find the distance to the top of the tree's shadow, we can visualize a right triangle formed by the height of the tree, the length of the shadow, and the distance from the base of the tree to the top of the shadow.
We know the following:
- Height of the tree (one side of the triangle) = 24 feet
- Length of the shadow (the other side of the triangle) = 10 feet
We need to determine the distance \( d \) to the top of the shadow, which will be the hypotenuse of the right triangle.
According to the Pythagorean theorem:
\[
d^2 = \text{(height)}^2 + \text{(shadow length)}^2
\]
Substituting the values we have:
\[
d^2 = 24^2 + 10^2
\]
\[
d^2 = 576 + 100
\]
\[
d^2 = 676
\]
Now, we take the square root of both sides to solve for \( d \):
\[
d = \sqrt{676}
\]
\[
d = 26 \text{ feet}
\]
Therefore, the distance to the top of the shadow is \( \boxed{26} \) feet.
We know the following:
- Height of the tree (one side of the triangle) = 24 feet
- Length of the shadow (the other side of the triangle) = 10 feet
We need to determine the distance \( d \) to the top of the shadow, which will be the hypotenuse of the right triangle.
According to the Pythagorean theorem:
\[
d^2 = \text{(height)}^2 + \text{(shadow length)}^2
\]
Substituting the values we have:
\[
d^2 = 24^2 + 10^2
\]
\[
d^2 = 576 + 100
\]
\[
d^2 = 676
\]
Now, we take the square root of both sides to solve for \( d \):
\[
d = \sqrt{676}
\]
\[
d = 26 \text{ feet}
\]
Therefore, the distance to the top of the shadow is \( \boxed{26} \) feet.