Question
For a ramp with a ramp length of 100 inches and an angle of 3 degrees find the horizontal run and the vertical rise. Below is your sketch, include the calculations you made to determine the horizontal run and vertical rise of your ramp. Round all final answers to the nearest tenth and input without units. Points: 3 points horizontal run with work shown. 3 points vertical rise with work shown. (2 points) Question 2 Show your work for the ramp vertical height and horizontal distance here. You may upload an image of your work if you prefer.(4 points)
Answers
GPT-4o mini
To determine the horizontal run and vertical rise of a ramp with a length of 100 inches and an angle of 3 degrees, we can use trigonometric functions.
### Definitions:
- **Ramp Length (hypotenuse)**: \( L = 100 \) inches
- **Angle of the ramp**: \( \theta = 3^\circ \)
### Vertical Rise Calculation:
The vertical rise can be calculated using the sine function, which relates the opposite side of a right triangle to the hypotenuse:
\[
\text{Vertical Rise} = L \cdot \sin(\theta)
\]
Substituting the known values:
\[
\text{Vertical Rise} = 100 \cdot \sin(3^\circ)
\]
Calculating \( \sin(3^\circ) \):
\[
\sin(3^\circ) \approx 0.05234
\]
Now, substituting this value back into our equation:
\[
\text{Vertical Rise} = 100 \cdot 0.05234 \approx 5.234
\]
Rounding to the nearest tenth:
\[
\text{Vertical Rise} \approx 5.2
\]
### Horizontal Run Calculation:
The horizontal run can be calculated using the cosine function, which relates the adjacent side of a right triangle to the hypotenuse:
\[
\text{Horizontal Run} = L \cdot \cos(\theta)
\]
Substituting the known values:
\[
\text{Horizontal Run} = 100 \cdot \cos(3^\circ)
\]
Calculating \( \cos(3^\circ) \):
\[
\cos(3^\circ) \approx 0.99863
\]
Now, substituting this value back into our equation:
\[
\text{Horizontal Run} = 100 \cdot 0.99863 \approx 99.863
\]
Rounding to the nearest tenth:
\[
\text{Horizontal Run} \approx 99.9
\]
### Summary of Results:
- **Horizontal Run**: 99.9
- **Vertical Rise**: 5.2
### Work Summary:
- **Vertical Rise Calculation**:
\[
Vertical\ Rise = 100 \cdot \sin(3^\circ) = 100 \cdot 0.05234 \approx 5.2
\]
- **Horizontal Run Calculation**:
\[
Horizontal\ Run = 100 \cdot \cos(3^\circ) = 100 \cdot 0.99863 \approx 99.9
\]
Feel free to ask if you require anything else!
### Definitions:
- **Ramp Length (hypotenuse)**: \( L = 100 \) inches
- **Angle of the ramp**: \( \theta = 3^\circ \)
### Vertical Rise Calculation:
The vertical rise can be calculated using the sine function, which relates the opposite side of a right triangle to the hypotenuse:
\[
\text{Vertical Rise} = L \cdot \sin(\theta)
\]
Substituting the known values:
\[
\text{Vertical Rise} = 100 \cdot \sin(3^\circ)
\]
Calculating \( \sin(3^\circ) \):
\[
\sin(3^\circ) \approx 0.05234
\]
Now, substituting this value back into our equation:
\[
\text{Vertical Rise} = 100 \cdot 0.05234 \approx 5.234
\]
Rounding to the nearest tenth:
\[
\text{Vertical Rise} \approx 5.2
\]
### Horizontal Run Calculation:
The horizontal run can be calculated using the cosine function, which relates the adjacent side of a right triangle to the hypotenuse:
\[
\text{Horizontal Run} = L \cdot \cos(\theta)
\]
Substituting the known values:
\[
\text{Horizontal Run} = 100 \cdot \cos(3^\circ)
\]
Calculating \( \cos(3^\circ) \):
\[
\cos(3^\circ) \approx 0.99863
\]
Now, substituting this value back into our equation:
\[
\text{Horizontal Run} = 100 \cdot 0.99863 \approx 99.863
\]
Rounding to the nearest tenth:
\[
\text{Horizontal Run} \approx 99.9
\]
### Summary of Results:
- **Horizontal Run**: 99.9
- **Vertical Rise**: 5.2
### Work Summary:
- **Vertical Rise Calculation**:
\[
Vertical\ Rise = 100 \cdot \sin(3^\circ) = 100 \cdot 0.05234 \approx 5.2
\]
- **Horizontal Run Calculation**:
\[
Horizontal\ Run = 100 \cdot \cos(3^\circ) = 100 \cdot 0.99863 \approx 99.9
\]
Feel free to ask if you require anything else!