A 40 foot high flagpole sits on the side of a hill. The hillside makes a 17 degree angle with horizontal. How long is a wire that runs from the top of the pole to a point 72 feet downhill from the base of the pole?
To find the length of the wire, we can use trigonometry and apply the concept of a right triangle. Here's a step-by-step explanation of how to solve the problem:
1. Visualize the problem: Draw a diagram of the situation described to better understand the scenario. Label the relevant measurements: the height of the flagpole (40 feet), the distance downhill from the base of the pole (72 feet), and the angle the hillside makes with the horizontal (17 degrees).
2. Identify the right triangle: From the diagram, you can see that the wire forms the hypotenuse of a right triangle, with the height of the flagpole as one of the other sides. The horizontal distance from the base of the pole to the point 72 feet downhill is the adjacent side, and the opposite side is the vertical distance from that point to the top of the pole.
3. Apply trigonometry: Since we have the opposite side (height of the pole) and the angle, we can use the sine function to find the length of the wire.
sin(angle) = opposite / hypotenuse
sin(17 degrees) = 40 feet / hypotenuse
4. Rearrange the equation to solve for the hypotenuse:
hypotenuse = opposite / sin(angle)
hypotenuse = 40 feet / sin(17 degrees)
5. Use a scientific calculator: Calculate the value of sin(17 degrees) either using the sine function on a scientific calculator or an online calculator. Make sure your calculator is set to degrees mode.
sin(17 degrees) ≈ 0.29237
6. Substitute the values into the equation:
hypotenuse = 40 feet / 0.29237
7. Calculate the length of the wire:
hypotenuse ≈ 136.83 feet
Therefore, the length of the wire that runs from the top of the pole to a point 72 feet downhill from the base of the pole is approximately 136.83 feet.