math question?

Question

A hill slopes at angle of 15 degree with teh horizantal. from the base of the hill, the angle of elevation of a 600 ft tower at the top of the hill is 40 degree. how much rope would be required to be reached from teh top of tower to the bottom of the hill? round answers to the nearest foot

Answer 1

let d be the horizontal distance from the base of the hill to the point at that elevation below the tower and let h be the height of the hill.
h/d = tan 15°, (h+600)/d = tan 40°, so
h = d tan 15
h + 600 = d tan 40
600 = d(tan 40 - tan 15)
d = 1050.5
then
h = 1050.5(tan 15) = 281.5

now, if the rope weighs nothing, its length x satisfies
x² = d² + (h+600)²
x² = 1050.5² + 881.5²
x² = 1880592.5
x = 1371.3

in the worst case, the rope follows the line of the tower, 600 ft, and then the slope of the hill, s, where
s² = 1050.5² + 281.5²
s² = 1182792.5
s = 1087.6,
for a length of 1687.6.
depending on the weight of the rope per foot and its tensile strength, the answer will be somewhere between those 2, but much closer to the 1687.