Question

A box has a volume of 546cm3 and a surface area of 422cm2. What are the Length, Width and Height?

Answer 1

We know that the formula for the volume of a box is:

$V =lwh$

Where:
l = length
w = width
h = height

The formula of the surface area of a box is:
$SA=2(lw+lh+wh)$

Based on the given we know the following:
$546cm^{3} =lwh$

and

$422cm^{2}=2(lw+lh+wh)$

If we factor out the given volume, you can come up with 3 numbers that will satisfy the surface area formula:

$546=6*7*13$

Let:
l = 6cm
w = 7cm
h = 13cm
     
Now let's try this on the surface area formula:
$422cm^{2}=2(lw+lh+wh)$
$422cm^{2}=2((6cm*7cm)+(6cm*13cm)+(7cm*13cm))$
$422cm^{2}=2(42cm^{2}+79cm^{2}+91cm^{2})$
$422cm^{2}=2(211cm^{2})$
$422cm^{2}=422cm^{2}$

The answer is then:
l = 6cm
w = 7cm
h = 13cm

Any order would be fine.