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?

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.

To find the length of the rope required, we can break down the problem into two components: the vertical component and the horizontal component.

Step 1: Find the vertical component of the rope.
The vertical component can be found by using the sine of the angle of inclination. The angle of inclination is 15 degrees.

Vertical component = 40 feet * sin(15 degrees)

Step 2: Find the horizontal distance.
The horizontal distance can be found by calculating the difference in height between the top of the hill and the point 68 feet downhill from the base of the pole.

Horizontal distance = 68 feet

Step 3: Calculate the length of the rope.
The length of the rope is the hypotenuse of a right-angled triangle formed by the vertical and horizontal components.

Length of the rope = √(Vertical component^2 + Horizontal component^2)

Substituting the values from Step 1 and Step 2:

Length of the rope = √((40 feet * sin(15 degrees))^2 + (68 feet)^2)

Using a calculator, the length of the rope is approximately 73.87 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 need to break down the problem into smaller steps.

Step 1: Find the height of the hill.
Since the hill is inclined at an angle of 15 degrees to the horizontal, we can use trigonometry to find the height. The vertical component of the height is given by the equation: height = length of the pole * sine(angle).
In this case, the length of the pole is 40 feet, and the angle is 15 degrees.
height = 40 * sin(15°)

Step 2: Find the length of the hypotenuse.
The hypotenuse of the right triangle formed by the pole and the hill is equivalent to the length of the rope needed. This can be calculated using the Pythagorean Theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.
In this case, one side is the height of the hill (found in Step 1), and the other side is the distance downhill, which is given as 68 feet.
hypotenuse^2 = height^2 + downhill distance^2

Step 3: Calculate the length of the rope.
To find the length of the rope, we need to take the square root of the value obtained in Step 2.
rope length = √(hypotenuse^2)

By following these steps, we can find the length of the rope required to reach from the top of the pole to a point 68 feet downhill from the base of the pole.