A 30-foot ladder leans against a wall, forming a $76^{\circ}$ angle with the ground. We need to find the height the ladder reaches on the wall, rounded to the nearest tenth of a foot.

1. Problem Description

A 30-foot ladder leans against a wall, forming a $76^{\circ}$ angle with the ground. We need to find the height the ladder reaches on the wall, rounded to the nearest tenth of a foot.

2. Solution Steps

Let $x$ be the height the ladder reaches on the wall. We have a right triangle where the ladder is the hypotenuse (30 feet), the angle between the ladder and the ground is $76^{\circ}$, and the height the ladder reaches on the wall is the opposite side to the angle. We can use the sine function to relate these quantities:

$sin(\theta) = \frac{opposite}{hypotenuse}$

In this case, $\theta = 76^{\circ}$, the opposite side is $x$, and the hypotenuse is

3

0. So we have:

$sin(76^{\circ}) = \frac{x}{30}$

Multiplying both sides by 30, we get:

$x = 30 * sin(76^{\circ})$

Using a calculator, we find that $sin(76^{\circ}) \approx 0.9703$

Therefore,

$x \approx 30 * 0.9703$

$x \approx 29.109$

Rounding to the nearest tenth of a foot, we get $x \approx 29.1$ feet.

3. Final Answer

The ladder reaches approximately 29.1 feet on the wall.