To find the length of the rope, we will break this problem up into a few steps. First, find the horizontal distance from the top of the pole to the base of the pole. Then, find the total horizontal distance from the top of the pole to the point 68 feet downhill. Finally, use the Pythagorean theorem to find the length of the rope.
Step 1:
Since the hill is inclined 15 degrees to the horizontal, and the pole is standing vertically, we can use the sine function to find the horizontal distance from the top of the pole to the base of the pole.
sin(15°) = horizontal distance / 40
To solve for the horizontal distance, multiply both sides of the equation by 40:
horizontal distance = 40 * sin(15°) ≈ 10.35 feet
Step 2:
Now, we need to find the total horizontal distance from the top of the pole to the point 68 feet downhill. This is simply the sum of the horizontal distance found in step 1 and the 68 feet downhill:
total horizontal distance = 10.35 + 68 ≈ 78.35 feet
Step 3:
We can now use the Pythagorean theorem to find the length of the rope. Let's call the length of the rope R, the vertical distance from the top of the pole to the bottom V (which we know is 40 feet), and the total horizontal distance H (which we found to be approximately 78.35 feet).
R^2 = V^2 + H^2
R^2 = 40^2 + 78.35^2 ≈ 8019.72
To find R, we simply take the square root of both sides of the equation:
R = sqrt(8019.72) ≈ 89.55 feet
Thus, it will take a rope of approximately 89.55 feet to reach from the top of the pole to a point 68 feet downhill from the base of the pole.
A hill is inclined 15 degrees to the horizontal. A 40-foot pole stands at the top of the hill. How long a rope will it take to reach from the top of the pole to a point 68 feet downhill from the base of the pole?
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