Question
To get from the ship to the dock, Mae must walk up a ramp with an angle of elevation of 17 degrees and a height of 5.3 feet. To the nearest tenth, how long is the ramp?
Answers
GPT-4o mini
To find the length of the ramp (denoted as \( L \)), we can use the sine function, which relates the angle of elevation, the height, and the length of the ramp. The relevant relationship in a right triangle is given by:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
In this scenario:
- The angle \( \theta \) is 17 degrees.
- The height (opposite side) is 5.3 feet.
- The length of the ramp (hypotenuse) is what we are trying to find.
Rearranging the sine function to solve for the hypotenuse gives us:
\[
L = \frac{\text{opposite}}{\sin(\theta)} = \frac{5.3}{\sin(17^\circ)}
\]
Now we will compute the sine of 17 degrees and then substitute to find \( L \):
Using a calculator, we find:
\[
\sin(17^\circ) \approx 0.2924
\]
Now plug this value into the formula:
\[
L = \frac{5.3}{0.2924} \approx 18.1 \, \text{feet}
\]
Thus, the length of the ramp, to the nearest tenth, is:
\[
\boxed{18.1} \, \text{feet}
\]
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
In this scenario:
- The angle \( \theta \) is 17 degrees.
- The height (opposite side) is 5.3 feet.
- The length of the ramp (hypotenuse) is what we are trying to find.
Rearranging the sine function to solve for the hypotenuse gives us:
\[
L = \frac{\text{opposite}}{\sin(\theta)} = \frac{5.3}{\sin(17^\circ)}
\]
Now we will compute the sine of 17 degrees and then substitute to find \( L \):
Using a calculator, we find:
\[
\sin(17^\circ) \approx 0.2924
\]
Now plug this value into the formula:
\[
L = \frac{5.3}{0.2924} \approx 18.1 \, \text{feet}
\]
Thus, the length of the ramp, to the nearest tenth, is:
\[
\boxed{18.1} \, \text{feet}
\]