The volume of a rectangular box is 480 in a unit of volume
Given:
A rectangular box has faces of area 48, 60, and 80 in unit of area.
Question:
What is its volume?
The Process:
Let us say l = length, w = width, and h = height.
Step-1: Areas
The areas of the faces of a rectangular box are 48, 60, and 80. Therefore we can write the following equations:
Step-2: Volume
The formula of volume of a rectangular box:
[tex]\boxed{ \ V = lwh \ }[/tex]
Both sides are squared.
[tex]\boxed{ \ V^2 = (lwh)^2 \ }[/tex]
[tex]\boxed{ \ V^2 = l^2 \cdot w^2 \cdot h^2 \ }[/tex]
[tex]\boxed{ \ V^2 = l \cdot l \cdot w \cdot w \cdot h \cdot h \ }[/tex]
[tex]\boxed{ \ V^2 = l \cdot w \cdot l \cdot h \cdot w \cdot h \ }[/tex]
[tex]\boxed{ \ V^2 = (lw)(lh)(wh) \ }[/tex]
And now, we substitute the previous three equations.
[tex]\boxed{ \ V^2 = 48 \times 60 \times 80 \ }[/tex]
[tex]\boxed{ \ V^2 = (3 \times 16)(4 \times 3 \times 5)(16 \times 5) \ }[/tex]
[tex]\boxed{ \ V^2 = (3 \times 3)(4 \times 5 \times 5)(16 \times 16) \ }[/tex]
[tex]\boxed{ \ V^2 = 3^2 \times 2^2 \times 5^2 \times 16^2 \ }[/tex]
[tex]\boxed{ \ V^2 = (3 \cdot 2 \cdot 5 \cdot 16)^2 \ }[/tex]
Let us do square roots on both sides.
[tex]\boxed{ \ \sqrt{V^2} = \sqrt{(3 \cdot 2 \cdot 5 \cdot 16)^2} \ }[/tex]
[tex]\boxed{ \ V = 3 \cdot 2 \cdot 5 \cdot 16 \ }[/tex]
V = 480
Thus, the volume of a rectangular box is 480 in a unit of volume.
Based on all the steps above, remember this:
[tex]\boxed{\boxed{ \ V^2 = (lw)(lh)(wh) \ }}[/tex]
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A cuboid is a 3D shape. Another name is a rectangular prism. Unlike the cube, the sides of a cuboid are unequal lengths. However, the way to calculate the volume of cuboid and cube is the same, i.e., multiplying the three lengths of the sides. The other thing is to calculate the surface area by adding up all the sides.
