A conveyor belt carries supplies from the first floor to the second floor, which is 24 feet higher. The belt makes a 60° angle with the ground. How far do the supplies travel from one end of the conveyor belt to the other? Round your answer to the nearest foot. If the belt moves at 75 ft/min, how long, to the nearest tenth of a minute, does it take the supplies to move to the second floor?

1 answer

We can use trigonometry to solve this problem. Let's call the distance the supplies travel "d."

First, we need to find the length of the inclined segment of the conveyor belt. This is the side adjacent to the angle of 60°, and its length is the height difference between the first and second floors. Using the formula for the sine function:

sin(60°) = opposite/hypotenuse

We get:

sin(60°) = 24/hypotenuse

Multiplying both sides by the hypotenuse, we get:

hypotenuse = 24/sin(60°)

hypotenuse ≈ 27.7 feet

Now we can use the Pythagorean theorem to find the length of the conveyor belt itself. This is the hypotenuse of a right triangle with sides d (the distance the supplies travel) and 27.7 feet.

d² + 27.7² = hypotenuse²

d² + 766.29 = 766.29

d² = 766.29 - 27.7²

d² ≈ 557.15

d ≈ 23.6 feet

So the supplies travel about 23.6 feet from one end of the conveyor belt to the other.

To find how long it takes the supplies to move to the second floor, we can use the formula:

time = distance/speed

The distance is 27.7 feet (the length of the conveyor belt), and the speed is 75 ft/min.

time = 27.7/75

time ≈ 0.37 minutes or about 0.4 minutes (rounded to the nearest tenth)

So it takes the supplies about 0.4 minutes (or 24 seconds) to move to the second floor.