Question

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.)

Answer 1

`\boxed {\boxed {\sf 69.28 \ feet}}`

Explanation:

Assuming that the tree is perpendicular with the ground, we can use trigonometric ratios to find the height of the tree.

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.

`tan ({\theta)=(opposite)/(adjacent)`

Substitute the values in.

`tan(30)=(h)/(120)`

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.

`120*tan(30)=(h)/(120)*120`

`120*tan(30)=h`

`120*0.5773502692=h`

`69.2820323=h`

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

`69.28 \approx h`

The height of the tree is about 69.28 feet.