Question

The bottom of a ladder must be placed 3 feet from a wall. The ladder is 20 feet long. How far above the ground does the ladder touch the wall? Round your answer to the nearest tenth.

Answer 1

To determine how far above the ground the ladder touches the wall, we can use the Pythagorean theorem. The ladder, the distance from the wall to the base of the ladder, and the height at which the ladder touches the wall form a right triangle.

Let:
- \( a \) = height above the ground where the ladder touches the wall (unknown)
- \( b \) = distance from the bottom of the ladder to the wall = 3 feet
- \( c \) = length of the ladder = 20 feet

According to the Pythagorean theorem:
\[
a^2 + b^2 = c^2
\]

Substituting the values we have:
\[
a^2 + 3^2 = 20^2
\]

Calculating the squares:
\[
a^2 + 9 = 400
\]

Now, isolate \( a^2 \):
\[
a^2 = 400 - 9
\]
\[
a^2 = 391
\]

Now, take the square root of both sides to find \( a \):
\[
a = \sqrt{391} \approx 19.8
\]

Rounding this to the nearest tenth gives us:
\[
\boxed{19.8}
\]

Thus, the ladder touches the wall at a height of approximately 19.8 feet above the ground.