Question

The angle of depression from an airplane to the top of an air traffic control tower is 56°. If the tower is 320 feet tall and the airplane is flying at an altitude of 7,450 feet, how far away is the airplane from the control tower?

Answer 1

There are two possible answers depending on how you interpret the question. Please read the explanation below carefully and in full

Diagonal distance from top of tower to plane :

`\boxed{d= 12,571\;feet}`

Horizontal distance from control tower to plane:

`\boxed{h = 10571\;feet}`

Explanation:

Before explanation, I must say the question is confusing. Are they asking for the distance from the top of the tower or the horizontal distance from the location of the tower?

I am giving answers for both. You can ask your professor about this and answer with the right one.

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This question and explanation can best be understood by drawing a diagram with given information

First, look at the attached Image
A is the top of the control tower 320 feet above the ground

B is the location of the plane 7450' above ground level.

The point C which lies on the vertical line is at the same height from the ground as the top of the control tower i.e. 320 feet

So B

Points ABC form a right triangle with m∠ACB = 90°

m∠ABC = 56° (given)

Since the 3 angles of a triangle add up to 180°, we have

m∠BAC + 56 + 90 = 180°

m∠BAC = 180 - (56 + 90) = 34°

Diagonal distance from top of tower(d)

The objective is to find d.

Use the law of sines which states:

If a, b and c are the lengths of the legs of a triangle opposite to the angles A, B and C respectively; then the law of sines states:

`(a)/(\sin A) = (b)/(\sin B) = (c)/(\sin C)`

Looking at the various side lengths and angles and applying this law we get

`(7130)/(\sin 34^\circ) = (d)/(\sin 90^\circ)\\\\\textrm{Since }\sin 90^\circ } = 1\\\\(7130)/(\sin 34^\circ) = d\\\\d = 12,750.5194 = 12,751 \textrm{ feet rounded to nearest foot}\\\\`

Horizontal distance from tower

Use the law of sines again.

Without calculating d first, we can use

`(h)/(\sin 56^\circ) = (7130)/(\sin 34^\circ)\\\\`

`h = (7130 (\sin 56^\circ))/(\sin 34^\circ) = 10,570.659 = 10,571 \;feet`

Therefore the possible answers are

Diagonal distance from top of tower to plane :

`\boxed{d= 12,571\;feet}`

Horizontal distance from control tower to plane:

`\boxed{h = 10571\;feet}`

It is highly likely that they are looking for the horizontal distance but check with your prof or include both answers