Volume of cylinders and prisms?

Question

There are 12 cylindrical cans with diameter 10 cm and height 20 cm in the box. Find the volume of air left in the box. (use pi= 3.14) 

Answer 1

The mechanics of this problem are to calculate the volume of the box, calculate the volume of the 12 cylinders (cans) and then subtract the cylinder volume from the box volume. The problem states to use the value of 3.14 for pi.

Calculate the Dimensions of the Box.

There are 12 cylinders. Each cylinder has a diameter of 10 cm and a height of 20 cm. Arrange the cylinders in a rectangular configuration of 3 rows and 4 columns (other configurations are possible). So, the length of the box is the diameter of one cylinder times 4 which equals 40 cm. The width of the box is the diameter of one cylinder times 3 which is 30 cm. The height of the box is equal to the height of a cylinder which is 20 cm.
So, the volume of the box equals 40 x 30 x 20 which is 24,000 cm^3

Calculate the Volume of the Cylinders

All cylinders have the same dimension: diameter = 10 cm and height = 20 cm. The volume of a Right Cylinder is        V = pi * r^2* h, where r = the radius and h = the height.

Volume of one cylinder: V = 3.14 * 5^2 * 20

Volume = 1570 cm^3

Volume of 12 cylinders: 12 * 1570 = 18,840 cm^3

Calculate the Difference in Volume Between the Box and the 12 cylinders

24,000 – 18,840 = 5,160 cm^3

So, the answer is 5,160 cm^3

Answer 2

The volume of the box is 20 x 30 x 20 = 12000       This true whether the cans are arranged 6 x 1 or 2 x 3.

 

The volume of each can is 500π, or 1571

 

The six can have a volume of 6(1571), or 9425

 

The air space is 12000 - 9425, or 2575 cm³