Question

Payton is building a ramp up to her home. She knows the height of the ramp is 2.5 feet. If the angle of elevation of the ramp is 22°, how long does the ramp have to be?

a. 2.70 feet
b. 8.33 feet
c. 1.01 feet
d. 6.67 feet

Answer 1

To find the length of the ramp (the hypotenuse in a right triangle), we can use the sine function, which relates the angle of elevation, the height (opposite side), and the hypotenuse.

The sine of an angle in a right triangle is defined as:

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

In this case:

- The angle of elevation is \(22^\circ\).
- The height (opposite side) is \(2.5\) feet.
- The length of the ramp (hypotenuse) is what we need to find.

Using the formula for sine, we can rearrange it to solve for the hypotenuse:

\[
\sin(22^\circ) = \frac{2.5}{\text{hypotenuse}}
\]

Rearranging gives:

\[
\text{hypotenuse} = \frac{2.5}{\sin(22^\circ)}
\]

Now, we calculate \(\sin(22^\circ)\):

\[
\sin(22^\circ) \approx 0.3746
\]

Now, substituting this value into the equation:

\[
\text{hypotenuse} = \frac{2.5}{0.3746} \approx 6.67 \text{ feet}
\]

Therefore, the length of the ramp has to be approximately **6.67 feet**.

The correct answer is **d. 6.67 feet**.