A steep mountain is inclined 74 degree to the horizontal and

A steep mountain is inclined 74 degree to the horizontal and rises h = 3500 ft above the surrounding plain. A cable car is to be installed from a point d = 900 ft from the base to the top of the mountain, as shown. Find the shortest length of cable needed. (Round your answer to the nearest foot.) ft

Solution

Solving for the base of the triangle formed
by the top of the mountain, the base of the
mountain, and the point directly below the
top of the mountain:

sin(74)/3500 = sin(90-74)/base
sin(74)/3500 = sin(16)/base
base * sin(74) = 3500 * sin(16)
base = 3500 * sin(16) / sin(74)
base =1003.60885

Using pythagorean theorem to solve for the length
of the cable car:

cable^2 = (3500^2 +1003.60885^2)
cable = sqrt(3500^2 + 1003.60885^2)
cable = 3641.04803

Thus, the cable must be approximately 3641.04803 ft.