We know that the formula for the volume of a box is:
[tex]V =lwh[/tex]
Where:
l = length
w = width
h = height
The formula of the surface area of a box is:
[tex]SA=2(lw+lh+wh)[/tex]
Based on the given we know the following:
[tex]546cm^{3} =lwh[/tex]
and
[tex]422cm^{2}=2(lw+lh+wh)[/tex]
If we factor out the given volume, you can come up with 3 numbers that will satisfy the surface area formula:
[tex]546=6*7*13 [/tex]
Let:
l = 6cm
w = 7cm
h = 13cm
Now let's try this on the surface area formula:
[tex]422cm^{2}=2(lw+lh+wh)[/tex]
[tex]422cm^{2}=2((6cm*7cm)+(6cm*13cm)+(7cm*13cm))[/tex]
[tex]422cm^{2}=2(42cm^{2}+79cm^{2}+91cm^{2})[/tex]
[tex]422cm^{2}=2(211cm^{2})[/tex]
[tex]422cm^{2}=422cm^{2}[/tex]
The answer is then:
l = 6cm
w = 7cm
h = 13cm
Any order would be fine.
