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A 40 foot tower must be secured with a wire anchored 5 feet from the base of the tower. What angle will the wire make with the ground? Round your answer to the nearest tenth.

A $40$ foot tower must be secured with a wire anchored $5$ feet from the base of the tower. What angle will the wire make with the ground? Round your answer to the nearest tenth.

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Full solution

Q. A

40

foot tower must be secured with a wire anchored

5

feet from the base of the tower. What angle will the wire make with the ground? Round your answer to the nearest tenth.

Identify Triangle Formed: Identify the right triangle formed by the tower, the ground, and the wire. The tower forms a vertical line to the ground, the ground is the horizontal line, and the wire is the hypotenuse of the right triangle.
Use Pythagorean Theorem: Use the Pythagorean theorem to find the length of the wire (hypotenuse). However, this step is not necessary because we are not asked to find the length of the wire, but the angle it makes with the ground.
Apply Trigonometry: Use trigonometry to find the angle the wire makes with the ground. We can use the tangent function, which is the ratio of the opposite side to the adjacent side in a right triangle. The opposite side is the height of the tower ( $40$ feet), and the adjacent side is the distance from the base of the tower to where the wire is anchored ( $5$ feet). So, $\tan(\text{angle}) = \frac{\text{opposite}}{\text{adjacent}} = \frac{40}{5}$ .
Calculate Angle: Calculate the angle using the inverse tangent function. $\newline$ $\text{angle} = \arctan(\frac{40}{5})$ $\newline$ $\text{angle} = \arctan(8)$ $\newline$ Now we use a calculator to find the angle.
Round to Nearest Tenth: Round the answer to the nearest tenth. $\newline$ Using a calculator, we find that $\arctan(8)$ is approximately $82.87498365$ degrees. $\newline$ Rounded to the nearest tenth, the angle is $82.9$ degrees.

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