A conveyor belt carries supplies from the first floor to the second floor, which is 12 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? If the belt moves at 80 ft / min how long does it take the supplies to move to the second floor to the nearest tenth of a minute.
A. 17 ft ; 0.2 min
B. 35 ft ; 0.4 min
C. 42 ft ; 0.5 min
D. 52 ft ; 0.6 min
We can use trigonometry to solve this problem. Let's call the distance the supplies travel "x". Then we can set up a right triangle with the conveyor belt as the hypotenuse, and the ground and the vertical height of the second floor as the legs:
[asy]
pair A=(0,0), B=(43.3,25), C=(43.3,0);
draw(A--B--C--cycle);
draw(rightanglemark(A,C,B,60));
label("60°",C+(1,0));
label("$x$ ft",B/2);
label("12 ft",B-C/2);
label("$h$ ft",C/2);
[/asy]
Using the angle and the vertical leg, we can find the length of the hypotenuse:
$$\sin 60^\circ = \frac{h}{x} \quad \Rightarrow \quad x = \frac{h}{\sin 60^\circ} = \frac{12}{\sqrt{3}/2} = 24\sqrt{3} \approx 41.6 \text{ ft}$$
So the supplies travel about 41.6 feet from one end of the conveyor belt to the other. To find how long it takes for the supplies to move to the second floor, we can divide the distance by the speed of the belt:
$$\frac{41.6 \text{ ft}}{80 \text{ ft/min}} \approx 0.52 \text{ min} \approx 0.5\text{ min}$$
So the answer is (C) 42 ft ; 0.5 min.
To find the distance the supplies travel from one end of the conveyor belt to the other, we can use trigonometry.
First, let's calculate the vertical height the supplies need to travel. Given that the second floor is 12 feet higher than the first floor, and the angle with the ground is 60°, we can use the sine function to find the vertical distance:
sin(60°) = Opposite/Hypotenuse
sin(60°) = H/12
Rearranging the equation to solve for H:
H = 12 * sin(60°)
H ≈ 12 * 0.866
H ≈ 10.392 ft
So, the supplies travel approximately 10.392 ft vertically.
To find the total distance traveled by the supplies, we need to calculate the length of the conveyor belt. This can be found using the cosine function:
cos(60°) = Adjacent/Hypotenuse
cos(60°) = L/12
Rearranging the equation to solve for L:
L = 12 * cos(60°)
L ≈ 12 * 0.5
L ≈ 6 ft
So, the supplies travel approximately 6 ft horizontally.
Now we can calculate the total distance traveled using the Pythagorean theorem:
Distance = √(Horizontal distance)^2 + (Vertical distance)^2
Distance ≈ √(6^2 + 10.392^2)
Distance ≈ √(36 + 108)
Simplifying:
Distance ≈ √144
Distance ≈ 12 ft
Therefore, the supplies travel approximately 12 ft from one end of the conveyor belt to the other.
To find the time it takes for the supplies to move from the first floor to the second floor, we can use the equation:
Time = Distance / Speed
Given that the belt moves at 80 ft/min and the distance is 10.392 ft, we can plug in these values:
Time = 10.392 ft / 80 ft/min
Time ≈ 0.130 min
Rounding to the nearest tenth of a minute, the time it takes for the supplies to move from the first floor to the second floor is approximately 0.1 min (or 0.2 min after rounding up).
So, the answer is A) 17 ft; 0.2 min.