QuestionKristen's telescope is 5 ft high. At an $8^{\circ}$ angle, how tall is the tree 120 ft away?

Studdy Solution

STEP 1

Assumptions1. The height of the tripod is5 feet. . The angle of elevation to the top of the tree is $8^{\circ}$ .
3. The distance to the tree is120 feet.
4. We are assuming that the ground is level and the tree is perpendicular to the ground.

STEP 2

We can use the tangent of the angle of elevation to find the height of the tree. The tangent of an angle in a right triangle is equal to the opposite side (height of the tree) divided by the adjacent side (distance to the tree).
$\tan(\theta) = \frac{opposite}{adjacent}$

STEP 3

Plug in the given values for the angle of elevation and the distance to the tree.
$\tan(8^{\circ}) = \frac{height\, of\, tree}{120\, feet}$

STEP 4

Rearrange the equation to solve for the height of the tree.
$height\, of\, tree = \tan(8^{\circ}) \times120\, feet$

STEP 5

Calculate the height of the tree.
$height\, of\, tree = \tan(8^{\circ}) \times120\, feet \approx16.8\, feet$

STEP 6

The total height of the tree is the height measured from the top of the tripod plus the height of the tripod.
$Total\, height = height\, of\, tree + height\, of\, tripod$

STEP 7

Plug in the values for the height of the tree and the height of the tripod.
$Total\, height =16.\, feet +5\, feet$

STEP 8

Calculate the total height of the tree.
$Total\, height =16.8\, feet +5\, feet =21.8\, feet$ The tree is approximately21.8 feet tall.

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