Mathematics College

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?

Answers

Answer 1

Answer:

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 :

[tex]\boxed{d= 12,571\;feet}[/tex]

Horizontal distance from control tower to plane:
[tex]\boxed{h = 10571\;feet}[/tex]

Step-by-step 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:
[tex]\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}[/tex]

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

[tex]\dfrac{7130}{\sin 34^\circ} = \dfrac{d}{\sin 90^\circ}\\\\\textrm{Since }\sin 90^\circ } = 1\\\\\dfrac{7130}{\sin 34^\circ} = d\\\\d = 12,750.5194 = 12,751 \textrm{ feet rounded to nearest foot}\\\\[/tex]

Horizontal distance from tower

Use the law of sines again.

Without calculating d first, we can use

[tex]\dfrac{h}{\sin 56^\circ} = \dfrac{7130}{\sin 34^\circ}\\\\[/tex]

[tex]h = \dfrac{7130 (\sin 56^\circ)}{\sin 34^\circ} = 10,570.659 = 10,571 \;feet[/tex]

Therefore the possible answers are

Diagonal distance from top of tower to plane :
[tex]\boxed{d= 12,571\;feet}[/tex]

Horizontal distance from control tower to plane:
[tex]\boxed{h = 10571\;feet}[/tex]

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

The Angle Of Depression From An Airplane To The Top Of An Air Traffic Control Tower Is 56. If The Tower

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What’s the value of x solve the equation

-3x+4=2x-6

Answers

Answer: x=2

Step-by-step explanation:

−3x+4=2x−6

Subtract 2x from both sides.

−3x+4−2x=−6

Combine −3x and −2x to get −5x.

−5x+4=−6

Subtract 4 from both sides.

−5x=−6−4

Subtract 4 from −6 to get −10.

−5x=−10

Divide both sides by −5.x=−5−10

​Divide −10 by −5 to get 2

x=2

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