A communication tower is 34m tall. Jane stands somewhere in front of the tower to the right and measures the angle of elevation to be 70 degrees. Bill stands somewhere in front of the tower to the left and measures the angle of elevation to be 50 degrees. How far apart are Jane and Bill if the angle between Jane, the base of the tower and bill is 90 degrees. Round to the nearest meter.

Did you make your sketch?

On mine I let Jane's distance to the tower's base be x m, and Bill's distance to the tower be y m
There are two right-angled triangles
For Jane:
tan70° = 34/x
x = 34/tan70°

do the same step for Bill's distance

add x and y

To solve this problem, we can begin by visualizing the scenario described.

Let's denote the distance between Jane and Bill as 'd', and let's consider the triangle formed between Jane, the base of the tower, and Bill. This triangle is a right-angled triangle because the angle between Jane, the base of the tower, and Bill is given as 90 degrees.

Now, let's break down the problem into two separate triangles: one involving Jane, the base of the tower, and the top of the tower, and another involving Bill, the base of the tower, and the top of the tower.

For Jane's triangle:
We know that Jane measures the angle of elevation to be 70 degrees, and the height of the tower is 34 meters. Using the tangent function, we can determine the distance between Jane and the base of the tower.

tan(angle) = opposite/adjacent
tan(70) = 34/adjacent
adjacent = 34/tan(70)
adjacent ≈ 14.29 meters

For Bill's triangle:
Similar to Jane's triangle, we know that Bill measures the angle of elevation to be 50 degrees and the height of the tower is 34 meters. Using the same approach, we can find the distance between Bill and the base of the tower.

tan(angle) = opposite/adjacent
tan(50) = 34/adjacent
adjacent = 34/tan(50)
adjacent ≈ 44.55 meters

Now, to find the distance between Jane and Bill, we can use the Pythagorean theorem since we have two sides of a right-angled triangle.

By applying the Pythagorean theorem:
distance^2 = (adjacent_Jane + adjacent_Bill)^2 + (height_of_tower)^2
distance^2 = (14.29 + 44.55)^2 + 34^2
distance^2 ≈ 58.84^2 + 34^2
distance^2 ≈ 3450.17 + 1156
distance^2 ≈ 4606.17
distance ≈ √4606.17
distance ≈ 67.86 meters

Therefore, Jane and Bill are approximately 67.86 meters apart, rounded to the nearest meter.