Math / Trigonometry

QuestionIf a tree has a height of 165 feet, what would be the angle of elevation from level ground measured from 95 feet away? Round your answer to the nearest tenth of a degree.
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Studdy Solution

STEP 1

1. The tree is perpendicular to the level ground.
2. The distance from the point on the ground to the base of the tree is $95$ feet.
3. The height of the tree is $165$ feet.

STEP 2

1. Identify the right triangle and the trigonometric function to use.
2. Set up the equation using the tangent function.
3. Solve for the angle of elevation.
4. Round the angle to the nearest tenth of a degree.

STEP 3

Identify the right triangle formed by the tree, the ground, and the line of sight. The tree's height is the opposite side, and the distance from the tree is the adjacent side. Use the tangent function:
$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

STEP 4

Set up the equation using the tangent function:
$\tan(\theta) = \frac{165}{95}$

STEP 5

Solve for the angle $\theta$ by taking the arctangent of both sides:
$\theta = \tan^{-1}\left(\frac{165}{95}\right)$
Calculate $\theta$ using a calculator:
$\theta \approx \tan^{-1}(1.7368)$

STEP 6

Round the angle $\theta$ to the nearest tenth of a degree:
$\theta \approx 60.3^\circ$
The angle of elevation is:
$\boxed{60.3^\circ}$

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Studdy Solution

STEP 1

1. The tree is perpendicular to the level ground.
2. The distance from the point on the ground to the base of the tree is $95$ feet.
3. The height of the tree is $165$ feet.

STEP 2

1. Identify the right triangle and the trigonometric function to use.
2. Set up the equation using the tangent function.
3. Solve for the angle of elevation.
4. Round the angle to the nearest tenth of a degree.

STEP 3

Identify the right triangle formed by the tree, the ground, and the line of sight. The tree's height is the opposite side, and the distance from the tree is the adjacent side. Use the tangent function:
$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

STEP 4

Set up the equation using the tangent function:
$\tan(\theta) = \frac{165}{95}$

STEP 5

Solve for the angle $\theta$ by taking the arctangent of both sides:
$\theta = \tan^{-1}\left(\frac{165}{95}\right)$
Calculate $\theta$ using a calculator:
$\theta \approx \tan^{-1}(1.7368)$

STEP 6

Round the angle $\theta$ to the nearest tenth of a degree:
$\theta \approx 60.3^\circ$
The angle of elevation is:
$\boxed{60.3^\circ}$

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Studdy Solution

STEP 1

1. The tree is perpendicular to the level ground.2. The distance from the point on the ground to the base of the tree is 95 95 95 feet.3. The height of the tree is 165 165 165 feet.

STEP 2

1. Identify the right triangle and the trigonometric function to use.2. Set up the equation using the tangent function.3. Solve for the angle of elevation.4. Round the angle to the nearest tenth of a degree.

STEP 3

STEP 4

Set up the equation using the tangent function:tan⁡(θ)=16595 \tan(\theta) = \frac{165}{95} tan(θ)=95165​

STEP 5

Solve for the angle θ\thetaθ by taking the arctangent of both sides:θ=tan⁡−1(16595) \theta = \tan^{-1}\left(\frac{165}{95}\right) θ=tan−1(95165​)Calculate θ\thetaθ using a calculator:θ≈tan⁡−1(1.7368) \theta \approx \tan^{-1}(1.7368) θ≈tan−1(1.7368)

STEP 6

Round the angle θ\thetaθ to the nearest tenth of a degree:θ≈60.3∘ \theta \approx 60.3^\circ θ≈60.3∘ The angle of elevation is:60.3∘ \boxed{60.3^\circ} 60.3∘​

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1. The tree is perpendicular to the level ground.
2. The distance from the point on the ground to the base of the tree is $95$ feet.
3. The height of the tree is $165$ feet.

1. Identify the right triangle and the trigonometric function to use.
2. Set up the equation using the tangent function.
3. Solve for the angle of elevation.
4. Round the angle to the nearest tenth of a degree.

Set up the equation using the tangent function:
$\tan(\theta) = \frac{165}{95}$

Solve for the angle $\theta$ by taking the arctangent of both sides:
$\theta = \tan^{-1}\left(\frac{165}{95}\right)$
Calculate $\theta$ using a calculator:
$\theta \approx \tan^{-1}(1.7368)$

Round the angle $\theta$ to the nearest tenth of a degree:
$\theta \approx 60.3^\circ$
The angle of elevation is:
$\boxed{60.3^\circ}$