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. (1 point) ________ feet
Answers
GPT-4o mini
To find the distance from the top of the tree to the top of its shadow, we can visualize a right triangle where:
- The height of the tree (24 feet) is one leg of the triangle.
- The length of the shadow (10 feet) is the other leg of the triangle.
- The distance we want to find, from the top of the tree to the top of the shadow, is the hypotenuse.
We will use the Pythagorean theorem, which states:
\[
a^2 + b^2 = c^2
\]
where \(c\) is the hypotenuse, and \(a\) and \(b\) are the other two sides. In this case:
- \(a = 24\) (height of the tree)
- \(b = 10\) (length of the shadow)
Plugging in the values:
\[
24^2 + 10^2 = c^2
\]
Calculating the squares:
\[
576 + 100 = c^2
\]
Adding the two results:
\[
676 = c^2
\]
To find \(c\), we take the square root of both sides:
\[
c = \sqrt{676} = 26
\]
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) is one leg of the triangle.
- The length of the shadow (10 feet) is the other leg of the triangle.
- The distance we want to find, from the top of the tree to the top of the shadow, is the hypotenuse.
We will use the Pythagorean theorem, which states:
\[
a^2 + b^2 = c^2
\]
where \(c\) is the hypotenuse, and \(a\) and \(b\) are the other two sides. In this case:
- \(a = 24\) (height of the tree)
- \(b = 10\) (length of the shadow)
Plugging in the values:
\[
24^2 + 10^2 = c^2
\]
Calculating the squares:
\[
576 + 100 = c^2
\]
Adding the two results:
\[
676 = c^2
\]
To find \(c\), we take the square root of both sides:
\[
c = \sqrt{676} = 26
\]
Thus, the distance from the top of the tree to the top of its shadow is:
\[
\boxed{26} \text{ feet}
\]