Question

cool down
lesson 7: applying ratios in right triangles
cool down: tallest tree
the tallest tree in the world is a redwood in california (as of january 2024).
if youre standing on the trail 220 feet from the bottom of the tree, you have to look up at a 60° angle to see the top. how tall is the tree?

Explanation:

Step1: Define trigonometric relationship

We use the tangent function for the right triangle formed by the observer, the base of the tree, and the top of the tree. The tangent of the angle of elevation equals the tree height (opposite side) divided by the horizontal distance (adjacent side):
$\tan(\theta) = \frac{\text{height}}{\text{distance}}$

Step2: Rearrange to solve for height

Isolate the height by multiplying both sides by the horizontal distance:
$\text{height} = \text{distance} \times \tan(\theta)$

Step3: Substitute given values

The distance is 220 feet, and $\theta = 60^\circ$. We know $\tan(60^\circ) = \sqrt{3} \approx 1.732$:
$\text{height} = 220 \times \tan(60^\circ) = 220\sqrt{3}$

Answer:

$220\sqrt{3}$ feet (or approximately 381.05 feet)