8.
A conveyor belt carries supplies from the first floor to the second floor, which is 27 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?
47 ft; 38 min
31 ft; 0.4 min
38 ft; 23.3 min
16 ft; 1.1 min
You are just looking for the hypotenuse of a right-angled triangle with height 27 and base angle of 60°
using fundamental trig ....
sin60° = 27/h, where h is the hypotenuse
h = 27/sin60 = 31.1769 ft
= 31 ft, 2 inches
time =distance/rate = 31.1769/75 = .416 minutes
choose your answers
your neighbor is standing at the top of a 14 foot ladder. the base is 4 feet from the house. how far off the ground is your neighbor
To find the distance the supplies travel from one end of the conveyor belt to the other, we can use the trigonometric relationship of the angle and the vertical distance.
Given:
Height difference between floors = 27 feet
Angle of the conveyor belt with the ground = 60°
Since we know the angle and the height difference, we can use trigonometry to find the distance traveled.
Using the sine function:
sin(angle) = opposite/hypotenuse
In this case, the opposite side is the height difference between floors, and the hypotenuse is the distance traveled.
sin(60°) = 27 feet / hypotenuse
Rearranging the formula:
hypotenuse = 27 feet / sin(60°)
Using a calculator:
hypotenuse ≈ 31 feet
Therefore, the supplies travel approximately 31 feet from one end of the conveyor belt to the other.
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 conveyor belt.
Given:
Speed of the conveyor belt = 75 ft/min
Distance traveled = 31 feet
Time = Distance / Speed
Time = 31 feet / 75 ft/min
Using a calculator:
Time ≈ 0.4133 minutes
Rounding to the nearest tenth of a minute:
Time ≈ 0.4 minutes
Therefore, it takes approximately 0.4 minutes (or 0.4 * 60 = 24 seconds) for the supplies 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 break down the problem into two components: the horizontal distance and the vertical distance.
1. Horizontal Distance:
The horizontal distance is the distance along the ground. We can use the angle of 60° to determine the horizontal component of the conveyor belt's length. We can use the cosine function because it relates the adjacent side to the hypotenuse in a right triangle.
cos(60°) = adjacent/hypotenuse
cos(60°) = horizontal distance/total distance
The total distance is the unknown value we are looking for. Rearranging the equation, we get:
horizontal distance = cos(60°) * total distance
2. Vertical Distance:
The vertical distance is the increase in height as the supplies are transported from the first floor to the second floor. This height difference is given as 27 feet.
Using Pythagoras' theorem, we can relate the horizontal and vertical distances to the total distance:
total distance = square root of (horizontal distance^2 + vertical distance^2)
Now, let's calculate the values:
1. Horizontal Distance:
cos(60°) = adjacent/hypotenuse
cos(60°) = horizontal distance/total distance
horizontal distance = cos(60°) * total distance
horizontal distance = 0.5 * total distance
2. Vertical Distance:
vertical distance = 27 feet
total distance = square root of (horizontal distance^2 + vertical distance^2)
total distance = square root of ((0.5 * total distance)^2 + 27^2)
total distance = square root of (0.25 * total distance^2 + 729)
(total distance)^2 = 0.25 * total distance^2 + 729
total distance^2 - 0.25 * total distance^2 = 729
0.75 * total distance^2 = 729
total distance^2 = 729 / 0.75
total distance^2 = 972
total distance = square root of 972
total distance ≈ 31.1288 feet ≈ 31 feet (rounded to the nearest foot)
So, the supplies travel approximately 31 feet from one end of the conveyor belt to the other.
To find the time it takes for the supplies to move to the second floor, we can use the formula:
time = distance / speed
Given that the belt moves at 75 ft/min, we can calculate the time:
time = 31 feet / 75 ft/min
time ≈ 0.4133 min ≈ 0.4 min (rounded to the nearest tenth of a minute)
Therefore, the supplies take approximately 0.4 minutes (or 38 seconds) to move to the second floor.
So, the correct answer is:
31 ft; 0.4 min