Question

A 24​-foot ladder is placed against a vertical wall of a​ building, with the bottom of the ladder standing on level ground 13 feet from the base of the building. How high up the wall does the ladder​ reach?
Question content area bottom
Part 1
The ladder reaches
  
enter your response here feet up the wall.  ​(Round to the nearest​ hundredth.)

Answer 1

To determine how high the ladder reaches up the wall, we can use the Pythagorean theorem. According to the theorem:

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

where:
- \( a \) is the height the ladder reaches on the wall,
- \( b \) is the distance from the bottom of the ladder to the wall,
- \( c \) is the length of the ladder.

In this case:
- \( c = 24 \) feet (length of the ladder),
- \( b = 13 \) feet (distance from the base of the ladder to the wall).

We need to find \( a \):

\[
a^2 + 13^2 = 24^2
\]

Calculating \( 13^2 \) and \( 24^2 \):

\[
a^2 + 169 = 576
\]

Now, subtract \( 169 \) from both sides:

\[
a^2 = 576 - 169
\]

\[
a^2 = 407
\]

Now, take the square root of both sides to find \( a \):

\[
a = \sqrt{407} \approx 20.174
\]

Rounding to the nearest hundredth:

\[
a \approx 20.17 \text{ feet}
\]

Therefore, the ladder reaches approximately **20.17 feet** up the wall.