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Volume

The volume of a rectangular solid (a box) is the product of the length, width, and height. The surface area is the sum of the area of the six faces:

 

If the length, width, and height of a rectangular solid (a box) are the same, it is a cube. Its volume is the cube of one of its sides, and its surface area is the sum of the areas of the six faces:

 

Example
The volume of the cube is x and its surface area is x. What is the length of an edge of the cube?
A.   6
B.   10
C.   18
D.   36
E.    48

Let e be the length of an edge of the cube. Recall that the volume of a cube is e3 and its surface area is 6e2. Since we are given that both the volume and the surface area are x, these expressions are equal:
 
e3 = 6e2
e3 – 6e2 = 0
e2(e – 6) = 0
e2 = 0 or  e – 6 = 0
e = 0 or  e = 6
 
We reject e = 0 since in that case no cube would exist. Hence, e = 6 and the answer is (A).
 


The volume of a cylinder is
V = πr2h, and the lateral surface (excluding the top and bottom) is S = 2πrh, where r is the radius and h is the height:





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