A conveyor belt carries supplies from the first floor to the second floor, which is 39 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 to the nearest foot? If the belt moves at 70 ft./min., how long does it take the supplies to move to the second floor to the nearest tenth of a minute?
travel = 39 ft / sin(60º)
time = travel / (70 ft/min)
To find the distance the supplies travel from one end of the conveyor belt to the other, we can use trigonometry. Considering the angle of 60° that the belt makes with the ground, we can use the sine function.
The sine function is defined as the opposite side divided by the hypotenuse in a right-angled triangle. In this case, the opposite side is the vertical distance the supplies travel (39 feet), and the hypotenuse is the distance they travel along the conveyor belt (which we'll call "d").
Therefore, we have the equation: sin(60°) = 39/d.
To solve for "d", we rearrange the equation as follows: d = 39 / sin(60°).
Calculating sin(60°) gives us √3/2. Substituting this value into the equation, we get: d = 39 / (√3/2).
Simplifying further, we multiply the numerator and denominator by 2: d = (39 * 2) / √3 = 78 / √3.
To find an approximate value to the nearest foot, we evaluate this expression using a calculator, which gives us approximately 45 feet. Therefore, the supplies travel approximately 45 feet from one end of the conveyor belt to the other.
Moving on to the next part of the question, we are given that the belt moves at a speed of 70 ft./min. We can use this information to find the time it takes for the supplies to move from the first floor to the second floor.
Since the supplies travel a distance of 39 feet vertically to reach the second floor, we can divide this distance by the speed of the belt to get the time: 39 / 70.
Calculating this gives us approximately 0.5571 minutes.
Rounding this value to the nearest tenth of a minute, we find that it takes approximately 0.6 minutes for the supplies to move from the first floor 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.
We have a right triangle formed with the conveyor belt being the hypotenuse. The angle between the ground and the conveyor belt is 60°, and the vertical side of the triangle is the height from the first floor to the second floor, which is 39 feet.
Using the trigonometric function sine (sin), we can find the length of the conveyor belt (opposite side) using the height (vertical side) and the angle (60°):
sin(60°) = opposite / hypotenuse
sin(60°) = 39 / hypotenuse
To isolate the hypotenuse:
hypotenuse = 39 / sin(60°)
Using a calculator, we find that sin(60°) is approximately 0.866.
hypotenuse = 39 / 0.866
hypotenuse ≈ 45
Therefore, the supplies travel approximately 45 feet from one end of the conveyor belt to the other to the nearest foot.
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 39 feet, and the speed is given as 70 ft./min.
time = 39 / 70
Using a calculator, we find that 39 / 70 is approximately 0.557, which when rounded to the nearest tenth, is 0.6.
Therefore, it takes approximately 0.6 minutes for the supplies to move to the second floor to the nearest tenth of a minute.