Question
For a ramp with a vertical rise of 15 inches and an angle of inclination of 4 degrees find the horizontal run, and ramp length. Include the calculations you made to determine the horizontal run and ramp length of your ramp. Round all final answers to the nearest tenth and input without units. Points: 3 points ramp length with work shown. 3 points horizontal run with work shown.
Answers
GPT-4o mini
To calculate the horizontal run and ramp length for a ramp with a vertical rise of 15 inches and an angle of inclination of 4 degrees, we can use trigonometry.
### Step 1: Find the Horizontal Run
Using the sine of the angle, we can find the horizontal run (base) using the formula:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
Where:
- \(\theta\) = angle of inclination (4 degrees)
- opposite = vertical rise (15 inches)
- hypotenuse = ramp length (not needed for this calculation)
To find the horizontal run (adjacent side), we can use the relationship of sine and cosine:
\[
\sin(\theta) = \frac{\text{vertical rise}}{\text{ramp length}}
\]
1. Rearranging for ramp length, we have:
\[
\text{ramp length} = \frac{\text{vertical rise}}{\sin(\theta)}
\]
2. To find the horizontal run, we use the cosine:
\[
\cos(\theta) = \frac{\text{horizontal run}}{\text{ramp length}}
\]
So,
\[
\text{horizontal run} = \text{ramp length} \cdot \cos(\theta)
\]
However, for now, let's calculate the ramp length using just the vertical rise and angle of inclination.
### Calculate Ramp Length First:
1. Calculate \( \sin(4^\circ) \):
\[
\sin(4^\circ) \approx 0.06976
\]
2. Find ramp length:
\[
\text{ramp length} = \frac{15}{\sin(4^\circ)} = \frac{15}{0.06976} \approx 215.6 \text{ inches}
\]
### Step 2: Find Horizontal Run
1. Calculate \( \cos(4^\circ) \):
\[
\cos(4^\circ) \approx 0.99756
\]
2. Find horizontal run:
\[
\text{horizontal run} = \text{ramp length} \cdot \cos(4^\circ)
\]
\[
\text{horizontal run} = 215.6 \cdot 0.99756 \approx 215.0 \text{ inches}
\]
### Final Answers:
- Horizontal run: **215.0**
- Ramp length: **215.6**
### Summary of Calculations:
- Ramp Length:
\[
\frac{15}{\sin(4^\circ)} \approx 215.6
\]
- Horizontal Run:
\[
215.6 \cdot \cos(4^\circ) \approx 215.0
\]
### Values (to the nearest tenth):
- Ramp Length: 215.6
- Horizontal Run: 215.0
### Step 1: Find the Horizontal Run
Using the sine of the angle, we can find the horizontal run (base) using the formula:
\[
\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}
\]
Where:
- \(\theta\) = angle of inclination (4 degrees)
- opposite = vertical rise (15 inches)
- hypotenuse = ramp length (not needed for this calculation)
To find the horizontal run (adjacent side), we can use the relationship of sine and cosine:
\[
\sin(\theta) = \frac{\text{vertical rise}}{\text{ramp length}}
\]
1. Rearranging for ramp length, we have:
\[
\text{ramp length} = \frac{\text{vertical rise}}{\sin(\theta)}
\]
2. To find the horizontal run, we use the cosine:
\[
\cos(\theta) = \frac{\text{horizontal run}}{\text{ramp length}}
\]
So,
\[
\text{horizontal run} = \text{ramp length} \cdot \cos(\theta)
\]
However, for now, let's calculate the ramp length using just the vertical rise and angle of inclination.
### Calculate Ramp Length First:
1. Calculate \( \sin(4^\circ) \):
\[
\sin(4^\circ) \approx 0.06976
\]
2. Find ramp length:
\[
\text{ramp length} = \frac{15}{\sin(4^\circ)} = \frac{15}{0.06976} \approx 215.6 \text{ inches}
\]
### Step 2: Find Horizontal Run
1. Calculate \( \cos(4^\circ) \):
\[
\cos(4^\circ) \approx 0.99756
\]
2. Find horizontal run:
\[
\text{horizontal run} = \text{ramp length} \cdot \cos(4^\circ)
\]
\[
\text{horizontal run} = 215.6 \cdot 0.99756 \approx 215.0 \text{ inches}
\]
### Final Answers:
- Horizontal run: **215.0**
- Ramp length: **215.6**
### Summary of Calculations:
- Ramp Length:
\[
\frac{15}{\sin(4^\circ)} \approx 215.6
\]
- Horizontal Run:
\[
215.6 \cdot \cos(4^\circ) \approx 215.0
\]
### Values (to the nearest tenth):
- Ramp Length: 215.6
- Horizontal Run: 215.0