From a point on the ground that is 120 feet from the base of a vertical tree, the angle of elevation to the top of the tree is 30°. Find the approximate height h of the tree to the nearest whole number of feet. (Round your answer to two decimal places.)

First, let's draw a diagram. From the point on the ground to the base, it is 120 feet and forms a 30 degree angle. We want to find the height of the tree, which is labeled h. (The diagram is attached and not to scale).

Next, recall the ratios.

sin(θ)= opposite/hypotenuse
cos(θ)= adjacent/hypotenuse
tan(θ)= opposite/adjacent

We see that the height is opposite the 30 degree angle and 120 is adjacent.

opposite=h
adjacent=120

Since we are given opposite and adjacent, we must use tangent.

[tex]tan ({\theta)=\frac{opposite}{adjacent}[/tex]

Substitute the values in.

[tex]tan(30)=\frac{h}{120}[/tex]

We are solving for h, so we must isolate it. It is being divided by 120 and the inverse of division is multiplication. Multiply both sides by 120.

[tex]120*tan(30)=\frac{h}{120}*120[/tex]

[tex]120*tan(30)=h[/tex]

[tex]120*0.5773502692=h[/tex]

[tex]69.2820323=h[/tex]

Round to the hundredth place (2 decimal places). The 2 in the thousandth place tells us to leave the 8 in the hundredth place.

[tex]69.28 \approx h[/tex]

The height of the tree is about 69.28 feet.