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Math  /  Geometry

Questionepression of a point 100 mts from the foot of the tree is 60∘60^{\circ}. Find the height of the tree.

Studdy Solution

STEP 1

1. The point is 100 meters away from the base of the tree, forming the adjacent side of a right triangle.
2. The angle of elevation to the top of the tree from this point is 60∘60^\circ.
3. The height of the tree forms the opposite side of the right triangle.
4. Trigonometric relationships can be used to solve for the height of the tree.

STEP 2

1. Identify the trigonometric function relating the angle of elevation, the distance to the tree, and the height of the tree.
2. Set up the equation involving the tangent function.
3. Solve the equation for the height of the tree.

STEP 3

Identify the trigonometric function that relates the given angle of elevation (60∘60^\circ), the distance to the tree (adjacent side, 100 meters), and the height of the tree (opposite side). The tangent function is appropriate here.
tan⁡(θ)=oppositeadjacent \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}

STEP 4

Set up the equation using the tangent function where θ=60∘\theta = 60^\circ, the height of the tree is the opposite side, and the distance from the point to the tree is the adjacent side.
tan⁡(60∘)=h100 \tan(60^\circ) = \frac{h}{100}

STEP 5

Recall that tan⁡(60∘)=3\tan(60^\circ) = \sqrt{3}. Substitute this value into the equation.
3=h100 \sqrt{3} = \frac{h}{100}

STEP 6

Solve for the height hh by multiplying both sides of the equation by 100.
h=100⋅3 h = 100 \cdot \sqrt{3}

STEP 7

Simplify the expression to find the numerical value of the height.
h=1003≈173.21 meters h = 100\sqrt{3} \approx 173.21 \ \text{meters}
The height of the tree is approximately 173.21173.21 meters.

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