Steel loading ramps are used to load a lawn mower onto a truck bed that is 37.5 inches above the ground. If each ramp makes a 30° angle with the ground, find the length of the ramp in inches.

We are given the height of the truck bed, which is the side opposite the angle of elevation, and the angle the ramp makes with the ground. We need to find the length of the ramp, which is the hypotenuse of the right-angled triangle formed.

• Opposite side (height) = 37.5 inches
• Angle ($\theta$) = 30°
• Hypotenuse (ramp length, $L$) = ?

The sine function relates the opposite side, the hypotenuse, and the angle:

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

We know that $\sin(30^\circ) = 0.5$.

$$0.5 = \frac{37.5}{L}$$

Rearrange the equation to solve for $L$:

$$L = \frac{37.5}{0.5}$$ $$L = 75$$

The length of the ramp is:

$$75\ \text{inches}$$