Question

For a ramp with an angle of elevation of 14° and a height of 2.5 feet, how long is the horizontal distance of the ramp? Round the answer to the nearest tenth. (1 point) The horizontal distance of the ramp is feet.

Answer 1

To find the horizontal distance of the ramp, we can use the sine function in a right triangle. The angle of elevation (14°) and the height (2.5 feet) provide us the necessary components to find the horizontal distance (adjacent side).

We can use the following trigonometric relationship:

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

Here, the opposite side is the height (2.5 feet) and the angle is 14°. To find the hypotenuse, we can rearrange and solve for the hypotenuse:

\[
\text{hypotenuse} = \frac{\text{opposite}}{\text{sin}(\theta)} = \frac{2.5}{\sin(14°)}
\]

Calculating \(\sin(14°)\):

\(\sin(14°) \approx 0.2419\)

Now, substituting back:

\[
\text{hypotenuse} \approx \frac{2.5}{0.2419} \approx 10.35 \text{ feet}
\]

Now, to find the horizontal distance (adjacent side), we can use the cosine function:

\[
\text{cos}(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\]

Rearranging to calculate the horizontal distance:

\[
\text{adjacent} = \text{hypotenuse} \cdot \text{cos}(\theta)
\]

Calculating \(\cos(14°)\):

\(\cos(14°) \approx 0.9703\)

Now substitute the hypotenuse value:

\[
\text{adjacent} \approx 10.35 \cdot 0.9703 \approx 10.04 \text{ feet}
\]

Rounding to the nearest tenth:

The horizontal distance of the ramp is approximately \(10.0\) feet.