QuestionKristen's telescope is 5 ft high, measuring an $8^{\circ}$ angle to a tree 120 ft away. Find the tree's height.

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 from the tripod to the tree is120 feet4. We are assuming a right triangle between the ground, the line of sight to the top of the tree, and the line from the telescope to the base of the tree5. We are looking for the total height of the tree

STEP 2

We can use the tangent of the angle of elevation to find the height of the tree above the telescope. The tangent of an angle in a right triangle is defined as the ratio of the opposite side (height of the tree above the telescope) to the adjacent side (distance from the telescope to the tree).
$\tan(\theta) = \frac{Opposite}{Adjacent}$

STEP 3

Substitute the given values into the equation.
$\tan(8^{\circ}) = \frac{Height_{tree}}{120}$

STEP 4

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

STEP 5

Calculate the height of the tree above the telescope.
$Height_{tree} = \tan(8^{\circ}) \times120 =16.77 \, feet$

STEP 6

The total height of the tree is the height of the tree above the telescope plus the height of the tripod.
$Total\, height = Height_{tree} + Height_{tripod}$

STEP 7

Substitute the given values into the equation.
$Total\, height =16.77\, feet +5\, feet$

STEP 8

Calculate the total height of the tree.
$Total\, height =16.77\, feet +5\, feet =21.77\, feet$ The tree is approximately21.77 feet tall.

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