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?
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.
To find the distance the supplies travel from one end of the conveyor belt to the other, we can use trigonometry.
Let's call the distance the supplies travel as "x". We can see that the conveyor belt forms a right triangle with the ground, where the angle between the ground and the belt is 60°. The vertical side of the triangle represents the 24 feet height difference between the two floors and the hypotenuse represents the distance the supplies travel.
Using the trigonometric function sine (sin), we can relate the opposite side (24 feet) to the hypotenuse (x) as follows:
sin(60°) = Opposite/Hypotenuse
sin(60°) = 24/x
We can rearrange this equation to solve for x:
x = 24 / sin(60°)
x ≈ 27.71
Therefore, the supplies travel approximately 27.71 feet from one end of the conveyor belt to the other (rounded to the nearest foot).
To find the time it takes for the supplies to move to the second floor, we need to divide the distance traveled by the speed of the belt.
Time = Distance / Speed
Time = 27.71 feet / 75 ft/min
Time ≈ 0.369
Therefore, it takes approximately 0.369 minutes (or 0.4 minutes when rounded to the nearest tenth) for the supplies to move to the second floor.