Kabir is flying a helicopter at an elevation of 1,350 feet. He sees an airplane 2,400 horizontal feet in front of him, but doesn't know how far the plane is above him. If the angle of elevation is 54 degrees, how far is the airplane from the ground?

Question

(1 point)
Responses

3,303 feet
3,303 feet

5,245 feet
5,245 feet

4,653 feet
4,653 feet

4,414 feet

Answer 1

To find the height of the airplane from the ground, we need to use the information given: the helicopter's elevation, the horizontal distance to the airplane, and the angle of elevation.

1. The helicopter's elevation is 1,350 feet.
2. The horizontal distance to the airplane is 2,400 feet.
3. The angle of elevation to the airplane is 54 degrees.

Let's denote the height of the airplane above the ground as \( h \).

Using the tangent of the angle of elevation, we can set up the following equation:

\[ \tan(54^\circ) = \frac{h - 1350}{2400} \]

Here, \( h - 1350 \) is the height of the airplane above the helicopter.

We can solve for \( h \):

1. Multiply both sides by 2400:

\[ 2400 \tan(54^\circ) = h - 1350 \]

2. Add 1350 to both sides:

\[ h = 2400 \tan(54^\circ) + 1350 \]

Now we can calculate \( \tan(54^\circ) \):

\[ \tan(54^\circ) \approx 1.37638 \]

Now substitute this value into the equation:

\[ h = 2400 \cdot 1.37638 + 1350 \] \[ h \approx 3304.032 + 1350 \] \[ h \approx 4654.032 \]

So, rounding gives approximately:

\[ h \approx 4,653 \text{ feet} \]

Thus, the height of the airplane from the ground is 4,653 feet.