A 52-foot wire running from the top of a tent pole to the ground makes an angle of 57° with the ground. If the length of the tent pole is 44 feet, how far is it from the bottom of the tent pole to the point where the wire is fastened to the ground? (The tent pole is not necessarily perpendicular to the ground. Enter your answers as a comma-separated list. Round your answers to the nearest whole number.)
To solve this problem, we can use trigonometry. Let's define the unknown distance from the bottom of the tent pole to the point where the wire is fastened as x.
Since the wire is running from the top of the tent pole to the ground, it forms a right triangle with the ground. The tent pole itself is not necessarily perpendicular to the ground, so we need to find the length of the adjacent side.
Using trigonometry, we can use the cosine function to relate the angle and the lengths of the sides:
cos(θ) = adjacent/hypotenuse
In this case, the adjacent side is x (the distance from the bottom of the tent pole to the point where the wire is fastened) and the hypotenuse is the length of the tent pole, which is given as 44 feet.
cos(57°) = x/44
To find the value of x, we can rearrange the equation:
x = cos(57°) * 44
Using a calculator, we can find:
x ≈ 23.64
Therefore, the distance from the bottom of the tent pole to the point where the wire is fastened is approximately 24 feet.
To solve this problem, we can use trigonometry. Let's start by sketching the situation described in the question:
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| \ 52ft
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|______\
44ft
In the diagram, the tent pole has a height of 44ft, and the wire is fastened to the top of the pole. The wire makes an angle of 57° with the ground.
We want to find the distance from the bottom of the tent pole to the point where the wire is fastened to the ground. Let's call this distance "x".
For a right triangle formed by the tent pole, the ground, and the wire, we have the following:
- The length of the side opposite the angle of 57° is x (the distance we want to find).
- The length of the side adjacent to the angle of 57° is 44ft (the height of the tent pole).
- The length of the hypotenuse (the wire) is given as 52ft.
To find "x", we can use the trigonometric function cosine (cos):
cos(angle) = adjacent / hypotenuse
In this case, the adjacent side is 44ft, and the angle is 57°:
cos(57°) = 44ft / 52ft
To solve for x, we can rearrange the equation:
x = cos(57°) * 52ft
Now, let's calculate the value of x:
x ≈ cos(57°) * 52ft
Using a scientific calculator or trigonometric table, we find that cos(57°) is approximately 0.5446:
x ≈ 0.5446 * 52ft
x ≈ 28.32ft
Rounding to the nearest whole number, the distance from the bottom of the tent pole to the point where the wire is fastened to the ground is approximately 28ft.
there is redundant (and slightly inconsistent) information
the distance is
52 cos57° = 28.32
√(52^2 - 44^2) = 27.71