SOLUTION: In the figure, a person standing at point A notices that the angle of elevation to the top of the antenna is 45° 30'. A second person standing 39.0 feet farther from the antenna t

Question 1207377: In the figure, a person standing at point A notices that the angle of elevation to the top of the antenna is 45° 30'. A second person standing 39.0 feet farther from the antenna than the person at A finds the angle of elevation to the top of the antenna to be 44° 10'. How far is the person at A from the base of the antenna? (Round your answer to the nearest whole number.)
Found 3 solutions by josgarithmetic, Theo, ikleyn:
Answer by josgarithmetic(39617)   (Show Source): You can put this solution on YOUR website!
Drawing of description gives two right triangles.

Larger triangle,
horizontal length x+39
angle on ground 44 degree 10 minute
antenna length y

Smaller triangle,
horizontal length x
angle on ground 45 degree 30 minute
antenna length y

You would or could use tangent of the angles, you want to find x, and y will be substituted and not be shown in single final equation to solve.
.
.

Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
angle A is equal to 45 degrees 30 minutes = 45 + 30/60 = 45.5 degrees.
angle B is equal to 44 degrees 10 minutes = 44 + 10/60 = 44.16666... degrees.

two triangle are formed with a common vertical side of h.
the angle closest to h is angle A.
the angle farthest to h is angle B.
angle A has an adjacent side of x and an opposite side of h.
angle B has an adjacent side of x + 39 and an opposite side of h.

you have tan(A) = h/x and you have tan(B) = h / (x + 39).

solve for h in both of those equations to get h = x * tan(A) and h = (x + 39) * tan(B).

simplify to get h = x * tan(A) and h = x * tan(B) + 39 * tan(B)

since h = h, then x * tan(A) = x * tan(B) + 39 * tan(B)

subtract x * tan(B) from both sides of the equation and factor out the x to get x * (tan(A) - tan(B)) = 39 * tan(B)

divide both sides of the equation by (tan(A) - tan(B)) to get x = 39 * tan(B) / (tan(A) - tan(B))

since A = 45.5 degrees and B = 44.166666..... degrees, you get x = 39 * tan(44.166666...) / (tan(45.5 - tan(44.166666...) = 818.5122291.

that's how far point A is from the base of the antenna.

i confirmed the value of x was correct, because:

h = tan(A) * x = tan(45.5) * 818.5122291 = 832.9240955 and .....
h = tan(B) * (x + 39) = tan(44.16666...) * 857.5122291 = 832.9240955.
they're the same value as they should be.

here's my diagram.

i used storage of ti-84 plus to hold the values.
those storage locations are A, B, C, D, E, and F indicated by ----> .....

let me know if you have any questions.
theo

Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.

If you want to see many other solved similar and different problems, look into the lesson

    - Find the height

in this site.   Learn the subject from there.

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