SOLUTION: A satellite orbiting the earth passes directly overhead at observation stations in Phoenix and Los Angles, 340 mi apart. At an instant when the satellite is between these two stati
Algebra.Com
Question 214350: A satellite orbiting the earth passes directly overhead at observation stations in Phoenix and Los Angles, 340 mi apart. At an instant when the satellite is between these two stations, its angle of elevation is simultaneously observed to be 61° at Phoenix and 70° at Los Angeles. How far is the satellite from Phoenix?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
If you can ignore the curvature of the earth, then the satellite forms a triangle with phoenix and Los Angeles.
-----
Let A = the satellite.
let B = Phoenix
let C = Los Angeles.
-----
BC = 340 miles equals the distance between Phoenix and LA.
-----
Drop a perpendicular from A to intersect with BC at point D.
This forms the line AD which is perpendicular to BC at D.
-----
You have 2 right triangles.
They are:
ABD and ACD
-----
let BD = x
let DC = y
you have:
x + y = 340 miles
-----
let AD = z
-----
z is the perpendicular from the satellite to the ground (line AD intersecting with line BC at point D).
-----
x is the distance from Phoenix to point D.
y is the distance from Los Angeles to point D.
-----
Since tangent is equal to opposite divided by adjacent, we have:
tan(B) = z/x
tan(C) = z/y
-----
This results in z = x*tan(B) and z = y*tan(C)
-----
Since they both equal to z, then they are both equal to each other, so:
x*tan(B) = y*tan(C)
-----
Since we know that x+y = 340, then y = 340-x
-----
We substitute in the equation we just created to get:
x*tan(B) = (340-x)*tan(C)
-----
We remove parentheses to get:
x*tan(B) = 340*tan(C) - x*tan(C)
-----
we add x*tan(C) to both sides of this equation to get:
x*tan(B) + x*tan(C) = 340*tan(C)
-----
we factor x on the left side of this equation to get:
x*(tan(B)+tan(C)) = 340*tan(C)
-----
We divide both sides of this equation by (tan(B)+tan(C)) to get:
x = 340*tan(C)/(tan(B)+tan(C))
-----
Since C = 70 degrees and B = 61 degrees, then this equation becomes:
x = 340*tan(70)/(tan(61)+tan(70))
-----
We use our calculator to get:
x = 205.2372088
This make y = 340 - x to get:
y = 134.7627912
-----
Since tan(B) = z/x, then z = x*tan(B)
Since B = 61 degrees and x = 205.2372088, then:
z = x*tan(61)
This calculates out to be:
z = 370.2577258
-----
Since tan(C) = z/y, then z = y*tan(C)
Since C = 70 degrees and y = 134.7627912, then:
z = y*tan(70)
This calculates out to be:
z = 370.2577258
-----
z calculated both ways is identical proving that our values for x and y are good.
-----
We're still not done though.
We need the distance from the Satellite to Phoenix.
That would be the hypotenuse of triangle ABD which is the line AB.
-----
let m = line AB.
-----
Sin(B) = opposite/hypotenuse = z/m
Cos(B) = adjacent/hypotenuse = x/m
-----
Either one will get us m.
-----
Using Sin(B) = z/m, we multiply both sides of this equation by m to get:
m*Sin(B) = z
We divide both sides of this equation by Sin(B) to get:
m = z/Sin(B)
Since we know the value of z and Sin(B), this calculates out to be:
m = 423.335677 miles
-----
Using Cos(B) = x/m, we multiply both sides of this equation by m to get:
m*Cos(B) = x
We divide both sides of this equation by Cos(B) to get:
m = x/Cos(B)
Since we know the value of x and Cos(B), this calculates out to be:
m = 423.335677 miles.
-----
The value of m is identical both ways we calculated it which is as it should be so we are good.
-----
The answer to your question is:
The satellite is 423.335677 miles from Phoenix.
-----
A picture of this problem can be found at the following website:
http://theo.x10hosting.com/
-----
Click on problem number 214350. If it's not there when you look, wait 30 minutes and try again.
-----
RELATED QUESTIONS
The path of a satellite orbiting the earth causes the satellite to pass directly over two (answered by ikleyn)
The path of a satellite orbiting the earth causes it to pass directly over two tracking... (answered by ankor@dixie-net.com)
The path of a satellite orbiting the earth causes the satellite to pass directly over two (answered by Cromlix)
The path of a satellite orbiting the earth causes it to pass directly over two tracking... (answered by lwsshak3)
The path of a satellite orbiting the earth causes the satellite to pass directly over two (answered by ankor@dixie-net.com)
The path of a satellite orbiting the earth causes it to pass directly over two tracking... (answered by josgarithmetic,hahamaster)
A satellite is orbiting Earth 980 km above Earth’s surface. A receiving
dish is located... (answered by Cromlix)
At noon, two tracking stations on Earth, 20 km apart, measure the angles of elevation of... (answered by ikleyn)
1. A satellite is orbiting Earth on a path described by x2+y2=45 000 000. A second... (answered by ikleyn)