Question

The height of a radio tower is 400 ​feet, and the ground on one side of the tower slopes upward at an angle of 10degrees. ​(a) How long should a guy wire be if it is to connect to the top of the tower and be secured at a point on the sloped side 190 feet from the base of the​ tower? ​(b) How long should a second guy wire be if it is to connect to the middle of the tower and be secured at a point 190 feet from the base on the flat​ side?

Answer 1

(a) 412 ft

(b) 276 ft

Explanation:

Consider the attached diagram.

(a) The internal angle of triangle RBT at B is 90° -10° = 80°. Since we know lengths RB and BT, we can find the length RT using the law of cosines:

RT² = RB² +BT² -2·RB·BT·cos(80°) = 190² +400² -2·190·400·cos(80°)

RT² ≈ 169,705.477

RT ≈ √169,705.477 ≈ 411.95

The guy wire to the hillside should be about 412 feet long.

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(b) The Pythagorean theorem can be used to find the shorter wire length.

LM² = LB² +MB²

LM = √(190² +200²) = √76,100

LM ≈ 275.86

The guy wire to the flat side should be about 276 feet long.