Mamadou leans a 18 -foot ladder against a wall so that it forms an angle of \( 75^{\circ} \) with the ground. What's the horizontal distance betwee he base of the ladder and the bottom of the wall? Round your answer to the nearest tenth of a foot if necessary.

To find the horizontal distance between the base of the ladder and the bottom of the wall, we can use trigonometry. Specifically, we can use the cosine function, which relates the angle, the adjacent side (the horizontal distance), and the hypotenuse (the length of the ladder). Given: - Length of the ladder (hypotenuse) = 18 feet - Angle with the ground = \( 75^{\circ} \) The cosine of the angle is given by the formula: \[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \] In this case, the adjacent side is the horizontal distance \( d \), and the hypotenuse is 18 feet. Therefore, we can write: \[ \cos(75^{\circ}) = \frac{d}{18} \] To find \( d \), we rearrange the equation: \[ d = 18 \cdot \cos(75^{\circ}) \] Now, we need to calculate \( \cos(75^{\circ}) \). Using a calculator: \[ \cos(75^{\circ}) \approx 0.2588 \] Now we can substitute this value back into the equation for \( d \): \[ d \approx 18 \cdot 0.2588 \approx 4.6534 \] Rounding to the nearest tenth, we find: \[ d \approx 4.7 \text{ feet} \] Thus, the horizontal distance between the base of the ladder and the bottom of the wall is approximately \( \boxed{4.7} \) feet.