QuestionA wire stretches from a pole's top to a point 18 ft from its base at a $52^{\circ}$ angle. Find the pole's height and wire length.

Studdy Solution

STEP 1

Assumptions1. The wire stretches from the top of a vertical pole to a point on level ground18 ft from the base of the pole. . The wire makes an angle of $52^{\circ}$ with the ground.
3. We are assuming the pole is perfectly vertical and the ground is perfectly horizontal, forming a right angle between them.
4. We are using the trigonometric functions sine, cosine, and tangent, which are defined for right triangles.

STEP 2

We can form a right triangle with the pole as one side, the ground as another side, and the wire as the hypotenuse. The angle between the ground and the wire is given as $52^{\circ}$ .

STEP 3

We can use the trigonometric function tangent, which is defined as the ratio of the opposite side to the adjacent side in a right triangle, to find the height of the pole.
$\tan(\theta) = \frac{Opposite\, side}{Adjacent\, side}$

STEP 4

Now, plug in the given values for the angle and the adjacent side (the distance from the base of the pole to the point on the ground) to find the height of the pole.
$\tan(52^{\circ}) = \frac{Height\, of\, the\, pole}{18\, ft}$

STEP 5

Rearrange the equation to solve for the height of the pole.
$Height\, of\, the\, pole = \tan(52^{\circ}) \times18\, ft$

STEP 6

Calculate the height of the pole.
$Height\, of\, the\, pole = \tan(52^{\circ}) \times18\, ft \approx24.03\, ft$

STEP 7

Now, we can use the trigonometric function cosine, which is defined as the ratio of the adjacent side to the hypotenuse in a right triangle, to find the length of the wire.
$\cos(\theta) = \frac{Adjacent\, side}{Hypotenuse}$

STEP 8

Now, plug in the given values for the angle and the adjacent side (the distance from the base of the pole to the point on the ground) to find the length of the wire.
$\cos(52^{\circ}) = \frac{18\, ft}{Length\, of\, the\, wire}$

STEP 9

Rearrange the equation to solve for the length of the wire.
$Length\, of\, the\, wire = \frac{18\, ft}{\cos(52^{\circ})}$

STEP 10

Calculate the length of the wire.
$Length\, of\, the\, wire = \frac{18\, ft}{\cos(52^{\circ})} \approx23.62\, ft$ The height of the pole is approximately24.03 ft and the length of the wire is approximately23.62 ft.

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