Question

The angle of elevation of the top of a radio mast from a point due east of it and 96 m away from its base is 30°. From another point, due west of the mast, the angle of elevation of the top is 60°. Calculate the distance of the second point from the base of the mast.​

Answer 1

Answer:

32 m

Step-by-step explanation:

Trigonometric ratios

$\sf \sin(\theta)=\dfrac{O}{H}\quad\cos(\theta)=\dfrac{A}{H}\quad\tan(\theta)=\dfrac{O}{A}$

where:

• $\theta$ is the angle
• O is the side opposite the angle
• A is the side adjacent the angle
• H is the hypotenuse (the side opposite the right angle)

Calculate the height of the radio mast using the tan trig ratio.

Given:

• $\theta$ = 30°
• O = h (height of radio mast)
• A = 96 m

$\implies \sf tan(30^{\circ})=\dfrac{ \sf h}{96}$

$\implies \sf h=96 \tan (30^{\circ})$

$\implies \sf h=\dfrac{96 \sqrt{3}}{3}$

$\implies \sf h=32 \sqrt{3}\:\:m$

To calculate the distance from the second point to the base of the mast, use the tan trig ratio:

Given:

• $\theta$ = 60°
• O = 32√3 m
• A = d

$\implies \sf tan(60^{\circ})=\dfrac{ \sf 32 \sqrt{3}}{d}$

$\implies \sf \sqrt{3}=\dfrac{ \sf 32 \sqrt{3}}{d}$

$\implies \sf d=\dfrac{ \sf 32 \sqrt{3}}{\sqrt{3}}$

$\implies \sf d=32\:\: m$