Marna is flying a helicopter at 1,350 feet and sees an airplane 2,400 horizontal feet in front of her, but is also unknown vertical distance above her. Marna knows the angle of elevation is 54 degrees. How far is the airplane from the ground? (1 point)

Question

Responses

3,003 feet
3,003 feet

4,653 feet
4,653 feet

4,600 feet
4,600 feet

4,654 feet

Answer 1

To find the height of the airplane from the ground, we can use trigonometry. We have the following information:

• Marna's altitude: 1,350 feet (height of the helicopter).
• Horizontal distance from Marna to the airplane: 2,400 feet.
• Angle of elevation from Marna to the airplane: 54 degrees.

Let \( h \) be the vertical distance from Marna to the airplane. Therefore, the altitude of the airplane from the ground would be \( 1,350 + h \).

Using the tangent function, we know that:

\[ \tan(54^\circ) = \frac{h}{2400} \]

We can rearrange this equation to solve for \( h \):

\[ h = 2400 \cdot \tan(54^\circ) \]

Now we calculate \( \tan(54^\circ) \). The approximate value of \( \tan(54^\circ) \) is about 1.376.

Now substitute that value into the equation:

\[ h = 2400 \cdot 1.376 \approx 3302.4 \text{ feet} \]

Now, to find the height of the airplane from the ground:

\[ \text{Height of airplane} = 1,350 + h \approx 1,350 + 3302.4 \approx 4,652.4 \text{ feet} \]

This rounds to approximately 4,654 feet.

Therefore, the height of the airplane from the ground is 4,654 feet.