SOLUTION: Thank you for the help: You are in a plane flying at an altitude of 900 feet. You see a statue at an angle of depression at 57 degrees. You also see a building at an angle of d

Question 1154926: Thank you for the help:
You are in a plane flying at an altitude of 900 feet. You see a statue at an angle of depression at 57 degrees. You also see a building at an angle of depression at 68 degrees. How far is the statue from the building?
Found 2 solutions by jim_thompson5910, MathTherapy:
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!

Here's one way to draw the diagram. The diagram is not to scale.

The points are defined as follows:
A = base of the statue
B = base of the building
C = point on the ground directly under point D
D = plane's location
E = point to help label the angles of depression

The lengths we'll work with are
CD = 900 (height)
AB = x (distance between statue and building)
BC = y

The given angles of depression are
angle ADE = 57 degrees (the upper angle shown in red)
angle BDE = 68 degrees (the upper angle shown in blue)

Because DE and AC are horizontal lines, they are parallel. So we know that the alternate interior angles are congruent. This makes,
angle DAC = 57 degrees (the lower angle shown in red)
angle DBC = 68 degrees (the lower angle shown in blue)

In other words,
angle ADE = angle DAC = 57 (red pair of angles)
angle BDE = angle DBC = 68 (blue pair of angles)
because they are alternate interior angle pairs.

For now, focus your attention on right triangle BCD.
Ignore points A and E. Ignore the 57 degree angle for now.
When we focus solely on triangle BCD, the label "angle DBC" can be simplified to the label "angle B". So B = 68.

We'll use the tangent ratio to solve for y.
Recall the tangent of an angle is the ratio of the opposite over adjacent.

tan(angle) = opposite/adjacent
tan(B) = CD/BC .... again we're only focusing on triangle BCD
tan(68) = 900/y
y*tan(68) = 900
y = 900/tan(68)
y = 363.623603251641 ... use a calculator here; value is approximate
Make sure your calculator is in degree mode.

Now we can use this to find x. We'll use another tangent ratio. This time, focus on triangle ACD. Ignore point B and its associated angle. Ignore point E.

tan(angle) = opposite/adjacent
tan(A) = CD/AC
tan(A) = CD/(AB+BC)
tan(57) = 900/(x+y)
tan(57) = 900/(x+363.623603251641)

Do a bit of algebra to solve for x. It might help to replace "tan(57)" with a variable such as z.

tan(57) = 900/(x+363.623603251641)
z = 900/(x+363.623603251641)
z(x+363.623603251641) = 900
zx+363.623603251641z = 900
zx = 900-363.623603251641z
x = (900-363.623603251641z)/z
x = (900-363.623603251641*tan(57))/(tan(57)) ... plug in z = tan(57)
x = 220.843230626119
I used a calculator for the last step. This value is approximate as well.

No rounding instructions were given, so I would ask what your teacher wants you to round to. For now I'll just round to the nearest whole number.

Answer: Approximately 221 feet

Edit: the tutor @MathTherapy has a much more elegant solution, so I would go with that method instead. Though the diagram is still handy to have.

Answer by MathTherapy(10557) (Show Source): You can put this solution on YOUR website!

You are in a plane flying at an altitude of 900 feet. You see a statue at an angle of depression at 57 degrees. You also see a building at an angle of depression at 68 degrees. How far is the statue from the building?

Distance (S) from statue to point DIRECTLY below plane:
Distance (B) from building to point DIRECTLY below plane:
Distance from statue to building:
RELATED QUESTIONS
Help please thank you. Mai is flying a plane at an altitude of 1600 ft. She sights a... (answered by Boreal)
It is a common experience to hear the sound of a low flying airplane, and look at the... (answered by ankor@dixie-net.com)
It is a common experience to hear the sound of a low flying airplane, and look at the... (answered by ankor@dixie-net.com)
An air traffic controller is tracking two planes. To start, Plane A is at an altitude of... (answered by Boreal)
Hi, It's word problems (again) mixed with angles. Please help: The angle of... (answered by nerdybill)
An airplane flying at a velocity of 900 km/h (W) travels 400 km west how long will the... (answered by TimothyLamb)
You are flying in an open plane at an altitude of 2304 feet and you drop a Coca Cola... (answered by rapaljer)
You are flying in an open plane at an altitude of 7056 feet and you drop a Coca Cola... (answered by jsmallt9)
11. You are stationed at a radar base and you observe an unidentified plane at an... (answered by Cromlix)