Let's solve the problem step-by-step to find the height of the ramp, also known as the rise, from the ground at its highest end:
1. Understand the Problem:
- We are given the length of the ramp (hypotenuse in a right triangle) as 31 inches.
- The angle of elevation from the ground is 19 degrees.
- We need to find the vertical rise (opposite side of the triangle) and round it to the nearest tenth of an inch.
2. Identify the Trigonometric Function to Use:
- Since we have the hypotenuse and we need to find the opposite side, we use the sine function. In a right triangle:
[tex]\[
\sin(\text{angle}) = \frac{\text{opposite}}{\text{hypotenuse}}
\][/tex]
3. Set Up the Equation:
- Let [tex]\( \text{rise} \)[/tex] be the height of the ramp.
- Here, the sine function will be:
[tex]\[
\sin(19^\circ) = \frac{\text{rise}}{31}
\][/tex]
4. Solve for Rise:
- Rearrange the formula to solve for the rise:
[tex]\[
\text{rise} = 31 \times \sin(19^\circ)
\][/tex]
5. Calculate the Rise:
- Using a calculator, find [tex]\( \sin(19^\circ) \)[/tex].
- Multiplying the sine of 19 degrees by 31 will give:
[tex]\[
\text{rise} \approx 10.1
\][/tex]
6. Round the Result:
- The exact calculated rise is approximately 10.1 inches when rounded to the nearest tenth.
So, at its highest end, the ramp will rise approximately 10.1 inches from the ground.
