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)

Answer 1

To determine the distance from the top of the tree to the top of the shadow, we can visualize this as a right triangle. The height of the tree forms one side (the vertical leg) of the triangle, and the length of the shadow forms the other side (the horizontal leg) of the triangle.

Given:
- Height of the tree = 24 feet (vertical leg)
- Length of the shadow = 10 feet (horizontal leg)

We can apply the Pythagorean Theorem, which states that in a right triangle:

\[
a^2 + b^2 = c^2
\]

where \(c\) is the hypotenuse (the distance from the top of the tree to the top of its shadow), and \(a\) and \(b\) are the lengths of the two legs of the triangle.

Here:
- \(a = 24\) feet (height of the tree)
- \(b = 10\) feet (length of the shadow)

Now, substituting into the Pythagorean theorem:

\[
c^2 = 24^2 + 10^2
\]

Calculating \(24^2\):

\[
24^2 = 576
\]

Calculating \(10^2\):

\[
10^2 = 100
\]

Now we add these results:

\[
c^2 = 576 + 100 = 676
\]

Next, we take the square root of both sides to find \(c\):

\[
c = \sqrt{676} = 26
\]

Thus, the distance from the top of the tree to the top of its shadow is \(26\) feet.