Question

A carpenter constructed a rectangular sandbox with a capacity of 10 cubic feet. If the carpenter were to make a similar sandbox twice as long, twice as wide, and twice as high as the first sandbox, what would be the capacity, in cubic feet, of the second sandbox?

Answer 1

Hint: Capacity refers to the volume of the rectangular sandbox, so the volume of the first box is given. We can assume its dimensions and equate it to the volume to get its value. Then we can make the changes in dimension and use the previously calculated value to find the volume of the second rectangular box.
As the box is rectangular, we will use the formula for volume of a cuboid (as 3-D figure of rectangle is cuboid).
Volume of cuboid = Length X Breadth X Height

Complete step-by-step answer:
Let the dimensions of the first rectangular box be:
Length = x feet
Breadth = y feet
Height = z feet
Volume of this rectangular box is given by substituting these values in the formula:
Volume of cuboid
= Length X Breadth X Height
= \[x{\text{ }}ft \times y{\text{ }}ft \times z{\text{ }}ft\]
= xyz $ f{t^3} $
So the volume of first rectangular box is xyz $ f{t^3} $ _______ (1)
Given volume of first rectangular box = 10 cubic feet _________ (2)
Both (1) and (2) are equal as these both are volumes of the first box
  $ \Rightarrow $ xyz = 10 $ f{t^3} $ _________ (3)
It is given that the dimensions of the second box are twice as that of the first, so the dimensions of the second box are:
Length = 2x feet (twice as long as the first box)
Breadth = 2y feet (twice as wide as the first box)
Height = 2z feet (twice as high as the first box)
Volume of this rectangular box is given by substituting these values in the formula:
Volume of cuboid
= Length X Breadth X Height
= \[2x{\text{ }}ft \times 2y{\text{ }}ft \times 2z{\text{ }}ft\]
= 8xyz $ f{t^3} $
But from the equation (3), It can be seen that the value of xyz is 10, so the volume of this cuboid will be:
(8 X 10) $ f{t^3} $ = 80 $ f{t^3} $
  $ \Rightarrow $ Required volume of the second box = 80 $ f{t^3} $
Therefore, the capacity, in cubic feet, of the second sandbox is 80 cubic feet (80 $ f{t^3} $ )

Note: As the volume initially was given in cubic feet, we knew that the dimensions of the rectangular box will be in the feet itself (because products of 3 quantities in unit feet make the resultant quantity of cubic feet).
Volume is always measured in cubic units be it cubic meter, centimeter or feet.