Question
Marna is flying a helicopter at 1,350 feet and sees an airplane 2,400 horizontal feet in front of her, but is also an unknown vertical distance above her. Marna knows the angle of elevation is 54 degrees. How far is the airplane from the ground?(1 point) Responses 4,653 feet 4,653 feet 4,714 feet 4,714 feet 3,303 feet 3,303 feet 5,245 feet
Answers
GPT-4o mini
To determine how far the airplane is from the ground, we can use the information provided to solve for the unknown vertical distance above Marna in the helicopter.
First, we need to set up the scenario. Let \( h \) be the vertical distance above Marna (the helicopter) to the airplane. The total altitude of the airplane will then be \( 1,350 + h \) feet.
The angle of elevation \( \theta \) is given as \( 54^\circ \), and the horizontal distance (adjacent side) from Marna to the airplane is \( 2,400 \) feet.
Using the tangent function, we have:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{2,400}
\]
Thus, we can write:
\[
h = 2,400 \tan(54^\circ)
\]
Now, we can calculate the value of \( h \):
\[
h = 2,400 \tan(54^\circ) \approx 2,400 \times 1.37638 \approx 3,303.33 \text{ feet}
\]
Now, we can find the total altitude of the airplane from the ground:
\[
\text{Altitude of airplane} = 1,350 + h = 1,350 + 3,303.33 = 4,653.33 \text{ feet}
\]
Rounding down for practical purposes, we get that the airplane is approximately 4,653 feet from the ground.
Thus, the correct response is: **4,653 feet**.
First, we need to set up the scenario. Let \( h \) be the vertical distance above Marna (the helicopter) to the airplane. The total altitude of the airplane will then be \( 1,350 + h \) feet.
The angle of elevation \( \theta \) is given as \( 54^\circ \), and the horizontal distance (adjacent side) from Marna to the airplane is \( 2,400 \) feet.
Using the tangent function, we have:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{2,400}
\]
Thus, we can write:
\[
h = 2,400 \tan(54^\circ)
\]
Now, we can calculate the value of \( h \):
\[
h = 2,400 \tan(54^\circ) \approx 2,400 \times 1.37638 \approx 3,303.33 \text{ feet}
\]
Now, we can find the total altitude of the airplane from the ground:
\[
\text{Altitude of airplane} = 1,350 + h = 1,350 + 3,303.33 = 4,653.33 \text{ feet}
\]
Rounding down for practical purposes, we get that the airplane is approximately 4,653 feet from the ground.
Thus, the correct response is: **4,653 feet**.