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A cargo container that is 120 feet long, 40 feet wide, and 30 feet tall is to be filled to maximum capacity with identical barrels that are right circular cylinders. The barrels have a radius of 2 feet, a height of 6 feet, and can be both stacked and aligned in perfect rows. Which of the following is the closest approximation of the total capacity, in cubic feet, of the barrels that are to be placed in the storage container?
(A) 19,000
(B) 101,000
(C) 115,000
(D) 144,000
(E) 450,000
(A) 19,000
(B) 101,000
(C) 115,000
(D) 144,000
(E) 450,000
10 Sep 2020, 23:08
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It works out pretty well because the Dimensions of the Box are Multiples of the Dimensions of Each Cylinder.
The Diameter of 1 Cylinder = 4 ft
Across the Length, we can put: 120/ 4 = 30 Cylinders to a Row
Across the Width, we can put: 40/4 = 10 Cylinders to a Column
30 * 10 = 300 Cylinders can be placed on the Bottom Stack.
Since each Cylinder is 6 ft in Height and the Height of the Box is 30 ft, we can Stack: 30/6 = 5 Levels of 300 Cylinders = 1500 Cylinders can be stacked in the Box
What is the Volume/Capacity of these 1500 Cylinders?
Volume of 1 Cylinder = (pi) * (2)^2 * 6 = 24(pi)
Volume of 1500 Cylinders = 1, 500 * (24 (pi)) = 36, 000 (pi)
Approximating (pi) as around 3 (which is an UNDERESTIMATE, so the Actual Answer should be a SLIGHTLY HIGHER VALUE)
36, 000 * 3 = 108 , 000
-B- is too low and -D- is too high
Answer -C-
The Diameter of 1 Cylinder = 4 ft
Across the Length, we can put: 120/ 4 = 30 Cylinders to a Row
Across the Width, we can put: 40/4 = 10 Cylinders to a Column
30 * 10 = 300 Cylinders can be placed on the Bottom Stack.
Since each Cylinder is 6 ft in Height and the Height of the Box is 30 ft, we can Stack: 30/6 = 5 Levels of 300 Cylinders = 1500 Cylinders can be stacked in the Box
What is the Volume/Capacity of these 1500 Cylinders?
Volume of 1 Cylinder = (pi) * (2)^2 * 6 = 24(pi)
Volume of 1500 Cylinders = 1, 500 * (24 (pi)) = 36, 000 (pi)
Approximating (pi) as around 3 (which is an UNDERESTIMATE, so the Actual Answer should be a SLIGHTLY HIGHER VALUE)
36, 000 * 3 = 108 , 000
-B- is too low and -D- is too high
Answer -C-
General Discussion
11 Sep 2020, 07:17
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If you put each cylinder perfectly in a rectangular box with height 6, and length and width 4 (the diameter of the cylinder), those boxes would fit perfectly into the cargo container with no unused space (because the height is divisible by 6, and the other dimensions by 4). A cylinder only takes up less space than a box that perfectly contains it because of the area outside of the circular circumference of the cylinder, but inside the square confines of the box. Since a circle inscribed in a square takes up π/4 of the area of that square, all of our cylinders will occupy π/4 of the total volume of the cargo container. If we estimate π/4 to be 3/4, we'll get an answer slightly too small, so the answer here will be slightly bigger than (3/4)(40)(30)(120) = 108,000, and C must be right.
