The car is in a distance of 832.14 metres from the barn.
The cliff is 62.5 ft high.
The kite is 36.64 ft high from the ground.
With trigonometric ratios which is concerned with the relationship of an angle of a right-angled triangle to ratios of two side lengths, we can derive the lengths which firm a right triangle for each questions considering the given angles of elevation and depression. The basic trigonometric ratios includes;
sine, cosine and tangent.
The distance of the car from the barn is calculated using tangent of angle 31° as follows:
tan 31° = 500/car distance from the barn {opposite/adjacent}
car distance from the barn = 500/tan31
car distance from the barn = 832.14.
The hight of the cliff using the tangent of the angle elevation 38° is:
hight of cliff = 80 ft × tan 38°
hight of cliff = 62.50 ft
The hight of the kite from the ground is the vertical distance from the hand of the person holding it and his height so using sine of angle 50° we have
height of kite from the kite = (40ft × sin 50°) + 6 ft
height of kite from the kite = 30.64ft + 6ft
height of kite from the kite = 36.64ft.
Therefore, using trigonometric ratios of tangent and sine, we derive the distance of the car from the barn to be 832.14 metres, the cliff is 62.5 ft high and the kite is 36.64 ft high from the ground.