Question
A 24-foot-tall tree casts a shadow that is 10 feet long. What is the distance from the top of the tree to the top of its shadow? Apply a Pythagorean triple to solve this problem..
Answers
GPT-4o mini
To find the distance from the top of the tree to the top of its shadow, we can visualize the problem as a right triangle. In this triangle:
- The height of the tree (24 feet) serves as one leg.
- The length of the shadow (10 feet) serves as the other leg.
- The distance from the top of the tree to the top of the shadow will be the hypotenuse.
We can use the Pythagorean theorem to find the hypotenuse (\(c\)):
\[
c^2 = a^2 + b^2
\]
where \(a\) is the height of the tree (24 feet) and \(b\) is the length of the shadow (10 feet).
First, we calculate \(a^2\) and \(b^2\):
\[
a^2 = 24^2 = 576,
\]
\[
b^2 = 10^2 = 100.
\]
Now, we add these values:
\[
c^2 = 576 + 100 = 676.
\]
Next, we take the square root of \(c^2\) to find \(c\):
\[
c = \sqrt{676} = 26.
\]
Thus, the distance from the top of the tree to the top of its shadow is:
\[
\boxed{26 \text{ feet}}.
\]
- The height of the tree (24 feet) serves as one leg.
- The length of the shadow (10 feet) serves as the other leg.
- The distance from the top of the tree to the top of the shadow will be the hypotenuse.
We can use the Pythagorean theorem to find the hypotenuse (\(c\)):
\[
c^2 = a^2 + b^2
\]
where \(a\) is the height of the tree (24 feet) and \(b\) is the length of the shadow (10 feet).
First, we calculate \(a^2\) and \(b^2\):
\[
a^2 = 24^2 = 576,
\]
\[
b^2 = 10^2 = 100.
\]
Now, we add these values:
\[
c^2 = 576 + 100 = 676.
\]
Next, we take the square root of \(c^2\) to find \(c\):
\[
c = \sqrt{676} = 26.
\]
Thus, the distance from the top of the tree to the top of its shadow is:
\[
\boxed{26 \text{ feet}}.
\]