To determine the vertical distance from the bottom of the door to the ground given the horizontal distance and the angle between the ground and the ramp, we will use trigonometry. Specifically, we will use the tangent (tan) function because it relates the angle to the opposite side (the vertical height we are looking for) and the adjacent side (the horizontal distance we have).
Here's the step-by-step solution:
1. Identify Known Values:
- Horizontal distance (adjacent side) = 120 inches
- Angle with the ground (θ) = 4°
2. Use Tangent Function:
The tangent of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side:
[tex]\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\][/tex]
Here, the opposite side is the vertical distance we need to find. Let's denote this distance as [tex]\( h \)[/tex].
3. Set Up the Equation:
[tex]\[
\tan(4°) = \frac{h}{120 \text{ inches}}
\][/tex]
4. Solve for [tex]\( h \)[/tex] (Vertical Distance):
[tex]\[
h = 120 \text{ inches} \times \tan(4°)
\][/tex]
5. Calculate [tex]\(\tan(4°)\)[/tex]:
Using a calculator, find the value of [tex]\(\tan(4°)\)[/tex]:
[tex]\[
\tan(4°) \approx 0.0699
\][/tex]
6. Compute [tex]\( h \)[/tex]:
[tex]\[
h = 120 \text{ inches} \times 0.0699 \approx 8.388 \text{ inches}
\][/tex]
7. Round the Result:
To the nearest tenth:
[tex]\[
h \approx 8.4 \text{ inches}
\][/tex]
So, the distance from the bottom of the door to the ground is approximately 8.4 inches.
