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?(1 point)
Responses

8.33 feet
8.33 feet

1.01 feet
1.01 feet

2.70 feet
2.70 feet

6.67 feet

Answer 1

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

The sine function is defined as:

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

In this case:
- \(\theta = 22^\circ\) (angle of elevation)
- \(\text{opposite} = 2.5 \text{ feet}\) (height of the ramp)
- \(\text{hypotenuse} = \text{length of the ramp}\)

Rearranging the formula gives us:

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

Substituting the values:

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

Using a calculator to find \(\sin(22^\circ)\):

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

Now calculate the length of the ramp:

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

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