Question

length of ramp is 20 ft. find height and base

Answer 1

Answer on Question#38772 - Math - Trigonometry

Question.

Length of ramp is 20 ft. find height and base.

Solution

A ramp is a flat supporting surface tilted at an angle, with one end higher than the other. The inclined plane's geometry is based on a right triangle. The horizontal leg length is called base, the vertical leg length is called height. (See the figure below.)

The length of ramp is given, i. e. $l = 20$ . Let $b$ and $h$ denote the base and the height, respectively. To determine the length $b$ of the base, we can use the cosine function.

$$\cos \alpha = \frac {b}{l}$$

Multiplying by $l$ and substituting 20 for $l$ we obtain

$$b = l \cos \alpha = 2 0 \cos \alpha .$$

We use the sine function to find the height

$$\sin \alpha = \frac {h}{l}$$

In a similar manner as above we obtain

$$h = l \sin \alpha = 2 0 \sin \alpha .$$

Because the angle $\alpha$ is not given, one can see that the solution of the problem is not unique. If we take $\alpha = 30{}^{\circ}$ , then we get

$$b = l \cos 3 0 {}^ {\circ} = 2 0 \cos 3 0 {}^ {\circ} = 2 0 \cdot 0, 8 6 6 0 = 1 7, 3 2 \mathrm {f t},$$ $$h = l \sin 3 0 {}^ {\circ} = 2 0 \sin 3 0 {}^ {\circ} = 2 0 \cdot 0, 5 = 1 0 \mathrm {f t}.$$

But if we take $\alpha = 45{}^{\circ}$ , then we obtain

$$b = l \cos 4 5 {}^ {\circ} = 2 0 \cos 4 5 {}^ {\circ} = 2 0 \cdot 0, 7 0 7 1 = 1 4, 1 4 2 \mathrm {f t},$$ $$h = l \sin 4 5 {}^ {\circ} = 2 0 \sin 4 5 {}^ {\circ} = 2 0 \cdot 0, 7 0 7 1 = 1 4, 1 4 2 \mathrm {f t}.$$

That is in the last case the height and base are equal.

Answers:

$$b = l \cos \alpha , h = l \sin \alpha$$