A ranger in fire tower A spots a fire at a direction of 40 d

A ranger in fire tower A spots a fire at a direction of 40 degree. A ranger in fire lower B, which is 28 miles velocity east of tower A, spots the same fire at a direction of 116 degree. How far from tower A is the fire? Solve the problem. A tower is supported by a guy wire 648 ft long. If the wire makes an angle of 42 degree with respect to the ground and the distance from the point where the wire is attached to the ground and the tower is 295 ft, how tall is the tower? Round your answer to the nearest tenth.

Solution

Let the distance between the tower A and the fire be L and distance between tower B and fire be D.

Then, we know that the horizontal distance is 28 miles.

by Geometry,

Lcos40^o - Dcos116^o = 28

=> 0.766L + 0.438D = 28 .......................[1]

Also, since tower A and tower B are directly in the east - west direction, therefore their vertical component of distances must match.

[this is because tower A does not bear any angle with tower B]

so, Lsin40^o = Dsin114^o  ..........................[2]

=> D = 0.7036L

substitute this in equation [1]

this will give : 0.766L + 0.438[0.7036L] = 28

0.766L + 0.3084L = 28

=> L = 26.06 miles.

so the fire is at a distance of 26.06 miles from tower A.

2] Use Pythagoras\' theorem

a² + b² = c²

here, a = 295 ft and c = 648 ft

so (295)² + b² = (648)²

=> b = [467856 - 87025]^1/2 = 576.96 ft = 577 ft.

So the height of the tower is 577 ft.