Question
A plane takes off at an angle of 78 degrees and covers a horizontal distance of 950 m. Find the distance the plane traveled as it was taking off.
A. 929.24 m
B. 197.52 m
C. 971.22 m
D. 4,569.25 m
A. 929.24 m
B. 197.52 m
C. 971.22 m
D. 4,569.25 m
Answers
GPT-4o mini
To find the distance the plane traveled during takeoff, we can use the horizontal distance and the angle of takeoff. The relationship can be described using trigonometry, specifically the cosine function, since we have the angle and the horizontal distance.
Let:
- \( d \) be the distance traveled by the plane during takeoff,
- \( d_{horizontal} \) be the horizontal distance (950 m),
- \( \theta \) be the angle of takeoff (78 degrees).
The cosine of the angle is related to the horizontal distance and the distance traveled as follows:
\[
d_{horizontal} = d \cdot \cos(\theta)
\]
We can rearrange this formula to find \( d \):
\[
d = \frac{d_{horizontal}}{\cos(\theta)}
\]
Substituting the values:
1. Convert the angle to radians if needed (not necessary for using cosine functions in calculators):
- However, since most calculators can handle degrees directly, we can use it in degrees.
2. Calculate \( \cos(78^\circ) \):
\[
\cos(78^\circ) \approx 0.2079
\]
3. Substitute this into the formula:
\[
d = \frac{950 \, \text{m}}{0.2079} \approx 4,573.36 \, \text{m}
\]
Since this value (4,573.36 m) does not match any of the options provided, let’s double-check the calculation:
Calculating \( d \) again:
1. Compute \( \cos(78^\circ) \):
\[
\cos(78^\circ) = 0.2079
\]
2. Substitute back into the formula:
\[
d \approx \frac{950}{0.2079} \approx 4,573.36 \, \text{m}
\]
Considering the choices given:
- A. 929.24 m
- B. 197.52 m
- C. 971.22 m
- D. 4,569.25 m
The closest answer is **D. 4,569.25 m**.
Let:
- \( d \) be the distance traveled by the plane during takeoff,
- \( d_{horizontal} \) be the horizontal distance (950 m),
- \( \theta \) be the angle of takeoff (78 degrees).
The cosine of the angle is related to the horizontal distance and the distance traveled as follows:
\[
d_{horizontal} = d \cdot \cos(\theta)
\]
We can rearrange this formula to find \( d \):
\[
d = \frac{d_{horizontal}}{\cos(\theta)}
\]
Substituting the values:
1. Convert the angle to radians if needed (not necessary for using cosine functions in calculators):
- However, since most calculators can handle degrees directly, we can use it in degrees.
2. Calculate \( \cos(78^\circ) \):
\[
\cos(78^\circ) \approx 0.2079
\]
3. Substitute this into the formula:
\[
d = \frac{950 \, \text{m}}{0.2079} \approx 4,573.36 \, \text{m}
\]
Since this value (4,573.36 m) does not match any of the options provided, let’s double-check the calculation:
Calculating \( d \) again:
1. Compute \( \cos(78^\circ) \):
\[
\cos(78^\circ) = 0.2079
\]
2. Substitute back into the formula:
\[
d \approx \frac{950}{0.2079} \approx 4,573.36 \, \text{m}
\]
Considering the choices given:
- A. 929.24 m
- B. 197.52 m
- C. 971.22 m
- D. 4,569.25 m
The closest answer is **D. 4,569.25 m**.