A satellite orbiting the earth passes directly overhead at observation station in Bengaluru and Shriharikota
Answers
satelite orbiting the earth passes directly in the following way
Step-by-step explanation:
If you can ignore the curvature of the earth, then the satellite forms a triangle with phoenix and Los Angeles.
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Let A = the satellite.
let B = Phoenix
let C = Los Angeles.
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BC = 340 miles equals the distance between Phoenix and LA.
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Drop a perpendicular from A to intersect with BC at point D.
This forms the line AD which is perpendicular to BC at D.
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You have 2 right triangles.
They are:
ABD and ACD
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let BD = x
let DC = y
you have:
x + y = 340 miles
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let AD = z
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z is the perpendicular from the satellite to the ground (line AD intersecting with line BC at point D).
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x is the distance from Phoenix to point D.
y is the distance from Los Angeles to point D.
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Since tangent is equal to opposite divided by adjacent, we have:
tan(B) = z/x
tan(C) = z/y
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This results in z = x*tan(B) and z = y*tan(C)
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Since they both equal to z, then they are both equal to each other, so:
x*tan(B) = y*tan(C)
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Since we know that x+y = 340, then y = 340-x
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We substitute in the equation we just created to get:
x*tan(B) = (340-x)*tan(C)
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We remove parentheses to get:
x*tan(B) = 340*tan(C) - x*tan(C)
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we add x*tan(C) to both sides of this equation to get:
x*tan(B) + x*tan(C) = 340*tan(C)
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we factor x on the left side of this equation to get:
x*(tan(B)+tan(C)) = 340*tan(C)
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We divide both sides of this equation by (tan(B)+tan(C)) to get:
x = 340*tan(C)/(tan(B)+tan(C))
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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
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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
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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
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z calculated both ways is identical proving that our values for x and y are good.
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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.
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let m = line AB.
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Sin(B) = opposite/hypotenuse = z/m
Cos(B) = adjacent/hypotenuse = x/m
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Either one will get us m.
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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
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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.
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The value of m is identical both ways we calculated it which is as it should be so we are good.
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The answer to your question is:
The satellite is 423.335677 miles from Phoenix.
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