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?
Ah, I see we're talking about poles and ropes now. Well, let me calculate that for you. To find the length of the rope, we can use some trigonometry. So, grab your popcorn and hold on tight!
First, we need to find the distance from the top of the pole to the point directly above the base of the pole. This can be found using the sine function.
sin(15 degrees) = opposite / hypotenuse.
The hypotenuse in this case is the length of the rope, and the opposite side is the distance from the top of the pole to the point above the base.
So, the length of the rope = sin(15 degrees) * 68 feet.
Now, let me do the math. *Calculating sounds*
Drumroll, please! The length of the rope you would need is approximately 17.461 feet! Ta-da!
Remember, this calculation assumes the rope is perfectly straight, and there are no other factors like friction or tensions from circus clowns attempting to climb it. Safety first, my friend!
To solve this problem, we can divide it into two parts: the vertical and horizontal components.
1. Vertical Component:
From the top of the pole to a point 68 feet downhill, there is a vertical distance that needs to be covered. Let's calculate that.
The vertical distance can be found using the sine function:
sin(15 degrees) = opposite/hypotenuse
opposite = sin(15 degrees) * hypotenuse
We know that the hypotenuse is the length of the rope and the opposite side is 68 feet. So, we have:
sin(15 degrees) = 68/hypotenuse
2. Horizontal Component:
Now, let's calculate the horizontal distance from the base of the pole to the point 68 feet downhill.
The horizontal distance is the adjacent side of the right triangle formed. We can use the cosine function to find it:
cos(15 degrees) = adjacent/hypotenuse
Since the adjacent side is the same as the horizontal distance and the hypotenuse is the length of the rope, we have:
cos(15 degrees) = horizontal distance/hypotenuse
3. Combining the Vertical and Horizontal Components:
Now, we can solve the two equations simultaneously to find the length of the rope.
To find the length of the rope, we need to equate the two equations:
68/hypotenuse = horizontal distance/hypotenuse
Cross-multiplying, we have:
68 = horizontal distance
So, the length of the rope needed to reach from the top of the pole to a point 68 feet downhill from the base of the pole is 68 feet.
To find the length of the rope needed to reach from the top of the pole to a point 68 feet downhill from the base of the pole, we can use trigonometry.
First, let's draw a diagram to visualize the situation. Here's what it would look like:
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Now, let me explain the steps to find the length of the rope:
Step 1: Identify the right triangle formed by the hill, the pole, and the rope. The angle of inclination of the hill is given as 15 degrees.
Step 2: Determine the length of the shorter side of the right triangle, which is the height of the pole. It is given as 40 feet.
Step 3: Find the length of the longer side of the right triangle, which represents the distance from the top of the pole to the point 68 feet downhill from the base of the pole. This side is the hypotenuse of the right triangle, which we need to find.
Step 4: Use the trigonometric function cosine (cos) to calculate the length of the hypotenuse. The cosine of an angle in a right triangle is equal to the adjacent side divided by the hypotenuse. In this case, the adjacent side is the distance from the top of the pole to the point 68 feet downhill from the base of the pole, and the hypotenuse is the length of the rope we want to find.
Step 5: Apply the cosine function to calculate the length of the hypotenuse. The formula is:
cos(angle) = adjacent side / hypotenuse
In this case, the angle is 15 degrees, and the adjacent side is 68 feet. Let's solve for the hypotenuse:
cos(15 degrees) = 68 / hypotenuse
Step 6: Rearrange the equation to solve for the hypotenuse:
hypotenuse = 68 / cos(15 degrees)
Step 7: Use a scientific calculator or an online calculator to find the value of cos(15 degrees) and then calculate the length of the hypotenuse:
hypotenuse ≈ 70.71 feet
So, the length of the rope needed to reach from the top of the pole to a point 68 feet downhill from the base of the pole is approximately 70.71 feet.