For the next three questions consider the following situation: Chad is designing

Question

For the next three questions consider the following situation: Chad is designing a
rectangular cereal box. His boss says that the box should be half as wide as it is tall,
and the depth should be one half of the width.
1/Determine a model which gives the surface area, S, of the cereal box in terms of its
height, h.
2/Determine a model for the volume of the cereal box in terms of its depth, d.
3/Determine a model for the surface area of the cereal box in terms of its width, w.

could someone explain what's going on here?

Explanation / Answer

I think it would be easiest if we first turned the word problem into some simple equations.

It tells us the box is half as wide as it is tall, so that means that the height (h) is double the length of the width (w). Thus, we know that h = 2w.

It also tells us that the depth is half of the width, so d = 0.5w.

Now we kind of have all the variables connected.

1. Now we need to find a model for the surface area. The surface area of a box is given by 2(dw+wh+hd), since you have to cover all six sides. We need to turn this equation into a form that only uses h. This just means lots of substitution.

h = 2w, so that means that w = 0.5h
our second equation is d = 0.5w, so d= 0.5(0.5h) or 0.25h

So now we just take our new relations and substitute them back in.
S = 2(dw+wh+hd)
= 2((0.25h)(0.5h)+(0.5h)(h)+(h)(0.25h))
=2(0.125h2+0.5h2+0.25h2)

= 0.25h2+h2+0.5h2 = 1.75h2

Now we have the surface area equation in terms of h!

All we did was find the correlation between all the dimensions, adn then we made sure all of THOSE were in terms of h before subtituting them back into the large equation. We'll do it again in part 2 with volume in terms of depth.

2. So we know that d = 0.5w, so that means that w = 2d.

We also know that h = 2w = 2(2d) = 4d

The volume of a box is given by = h(d)(w)

Which here is equal to (4d)(d)(2d) = 8d3

We did the exact same substiution thing - we just used the relationships we found earlier to generate very simple equations in terms of d.

3. Finally, a surface area formula in terms of w.

So d = 0.5w and h = 2w

Thus S = 2(dw+wh+hd)

   = 2(0.5w(w)+(w)(2w)+(2w)(0.5w))

= 2(0.5w2+2w2+w2) = w2+4w2+2w2 = 7w2

I hope this helped!

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