To solve this problem, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b).
In this case:
- The length of the ladder is the hypotenuse (c = 10 feet).
- The distance from the bottom of the ladder to the base of the tree is one leg of the triangle (a = 6 feet).
- The height the ladder reaches up the tree is the other leg of the triangle (b, what we are trying to find).
Using the Pythagorean theorem:
\[ c^2 = a^2 + b^2 \]
Substituting in the known values:
\[ 10^2 = 6^2 + b^2 \]
\[ 100 = 36 + b^2 \]
Now, solve for \( b^2 \):
\[ b^2 = 100 - 36 \]
\[ b^2 = 64 \]
Now take the square root of both sides:
\[ b = \sqrt{64} \]
\[ b = 8 \]
Therefore, the height that the ladder reaches up the tree is 8 feet.
So the correct response is:
8 feet.