Question
Fatima and Tia are loading a refrigerator into the back of a truck. They know they need the ramp to make a 50-degree angle with the ground to reach the truck bed, which is 42 inches from the ground. How long is the ramp?Option #1: sin 50°=42x=55 inches Option #2: cos 50°=42x=48 inches Option #3: sin 42°=x42=63 inches Option #4: cos 42°=42x=62 inches
Answers
GPT-4o mini
To determine the length of the ramp needed to reach the truck bed, we can use trigonometric functions. The ramp, ground, and refrigerator form a right triangle, where the ramp is the hypotenuse, the height of the truck bed (42 inches) is the opposite side, and the adjacent side can be calculated using either sine or cosine functions.
Given:
- Height (opposite side) = 42 inches
- Angle = 50 degrees
1. **Using Sine:**
The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the hypotenuse.
\[
\sin(50^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{42}{x}
\]
Rearranging gives:
\[
x = \frac{42}{\sin(50^\circ)}
\]
Calculating \( \sin(50^\circ) \) (using a calculator):
\[
\sin(50^\circ) \approx 0.766
\]
Therefore:
\[
x \approx \frac{42}{0.766} \approx 54.8 \text{ inches}
\]
2. **Using Cosine:**
Alternatively, we can also find the length of the ramp using the cosine function. The cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse.
The height can be determined as:
\[
\cos(50^\circ) = \frac{\text{adjacent}}{x}
\]
However, we need to use the sine to find the hypotenuse based on the opposite side (the height of the truck bed).
From the given options:
- **Option #1:** \( \sin 50^\circ = \frac{42}{x} \rightarrow x = 54.8 \text{ inches} \) (closely matches)
- **Option #2:** Incorrect as it represents \( \cos \) related to the same height.
- **Option #3:** and **Option #4:** Incorrect as they do not calculate the hypotenuse based on given height or angle correctly.
Thus, the correct approach, based on our calculations and the closest to practical values, shows **Option #1** is the most accurate in context to calculating the ramp length needed.
**Final Answer: Ramp length is approximately 55 inches.**
Given:
- Height (opposite side) = 42 inches
- Angle = 50 degrees
1. **Using Sine:**
The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the hypotenuse.
\[
\sin(50^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{42}{x}
\]
Rearranging gives:
\[
x = \frac{42}{\sin(50^\circ)}
\]
Calculating \( \sin(50^\circ) \) (using a calculator):
\[
\sin(50^\circ) \approx 0.766
\]
Therefore:
\[
x \approx \frac{42}{0.766} \approx 54.8 \text{ inches}
\]
2. **Using Cosine:**
Alternatively, we can also find the length of the ramp using the cosine function. The cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse.
The height can be determined as:
\[
\cos(50^\circ) = \frac{\text{adjacent}}{x}
\]
However, we need to use the sine to find the hypotenuse based on the opposite side (the height of the truck bed).
From the given options:
- **Option #1:** \( \sin 50^\circ = \frac{42}{x} \rightarrow x = 54.8 \text{ inches} \) (closely matches)
- **Option #2:** Incorrect as it represents \( \cos \) related to the same height.
- **Option #3:** and **Option #4:** Incorrect as they do not calculate the hypotenuse based on given height or angle correctly.
Thus, the correct approach, based on our calculations and the closest to practical values, shows **Option #1** is the most accurate in context to calculating the ramp length needed.
**Final Answer: Ramp length is approximately 55 inches.**