Math / Trigonometry

QuestionAssuming that a 360 -foot tall giant redwood grows vertically, if I walk a certain distance from the tree and measure the angle of elevation to the top of the tree to be $47^{\circ}$ , how far from the base of the tree am I?
Round your answer to four decimal places.
I am about Number $\square$ feet away from the base of the tree.

Studdy Solution

STEP 1

1. The tree is growing vertically and forms a right triangle with the ground and the line of sight.
2. The height of the tree is $360$ feet.
3. The angle of elevation from the point on the ground to the top of the tree is $47^\circ$ .

STEP 2

1. Identify the trigonometric relationship.
2. Set up the equation using the tangent function.
3. Solve for the distance from the base of the tree.

STEP 3

Identify the trigonometric relationship:
In a right triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side. Here, the opposite side is the height of the tree, and the adjacent side is the distance from the base of the tree.
$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

STEP 4

Set up the equation using the tangent function:
Let $d$ be the distance from the base of the tree. Then,
$\tan(47^\circ) = \frac{360}{d}$

STEP 5

Solve for the distance $d$ :
Rearrange the equation to solve for $d$ :
$d = \frac{360}{\tan(47^\circ)}$
Calculate $\tan(47^\circ)$ using a calculator:
$\tan(47^\circ) \approx 1.0724$
Substitute this value back into the equation:
$d = \frac{360}{1.0724} \approx 335.6644$
You are about $\boxed{335.6644}$ feet away from the base of the tree.

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