Question

he bottom of a ladder must be placed 5 ft. from a wall. The ladder is 12 feet long.


Based on the diagram, what is the distance in feet from the ground that the ladder reaches the wall? Round to the nearest tenth.


$\text{Basic}$
$x$$y$$x^2$$\sqrt{ }$$\frac{x}{ }$
$x\frac{ }{ }$
$x^{ }$$x_{ }$$\degree$$\left(\right)$$\abs{ }$$\pi$$\infty$

Answer 1

To find the distance from the ground that the ladder reaches the wall, we can use the Pythagorean theorem. In this scenario, we have a right triangle formed by the wall, the ground, and the ladder.

Let:
- \( L \) be the length of the ladder (hypotenuse) = 12 ft
- \( d \) be the distance from the wall to the bottom of the ladder (one side of the triangle) = 5 ft
- \( h \) be the height at which the ladder touches the wall (the other side of the triangle) = ?

According to the Pythagorean theorem:
\[
L^2 = d^2 + h^2
\]

Substituting in the known values:
\[
12^2 = 5^2 + h^2
\]
\[
144 = 25 + h^2
\]

To isolate \( h^2 \), we subtract 25 from both sides:
\[
h^2 = 144 - 25 = 119
\]

Now we take the square root of both sides to solve for \( h \):
\[
h = \sqrt{119}
\]

Calculating \( \sqrt{119} \):
\[
h \approx 10.9087
\]

Rounding this to the nearest tenth:
\[
h \approx 10.9
\]

Therefore, the distance from the ground that the ladder reaches the wall is approximately **10.9 feet**.