# How to find the maximum volume of a box with inversely proportional sides

This problem is from my brother's calculus class, so it is slightly over my head (I'm in pre-calc), but I am curious how to go about solving this problem.

Basically there is a rectangle with sides of $$15$$ units and $$9$$ units in length, and four squares with sides that have a length of $$x$$ are cut out of each corner.

If you fold up each of the sides so that it makes a 3-dimensional shape, how would you figure out the maximum volume of the shape?

Here's what I have so far:

$$V = (15-2x)(9-2x)x$$

$$V = 4x^3 - 48x^2 + 135x$$

That's about as far as I've gotten. Can anybody help?

• A couple of things to note: (1) in the question title, you're asking for the maximum (surface) area, but in the problem, you're looking at the volume. These will not necessarily have the same answer. (2) Double check your $V$ calculation; it should have $135x$. Nov 3, 2022 at 13:27
• Yeah you're right. I'll fix the issues Nov 3, 2022 at 13:46
• Now take a derivative and set to zero Nov 3, 2022 at 14:07

Since the derivative is the slope of the curve, the point is a maximum or a minimum when derivative equals zero You have, $$V = 4x^3 - 48x^2 + 135x\\ \dfrac{dV}{dx}=12x^2-96x+135\\ \dfrac{dV}{dx}=0 \Rightarrow 12x^2-96x+135 =0$$ Solving this, we get $$x\approx6.2,1.8$$ Substituting these values on $$V$$,(or by excluding $$6.2$$, since one of the sides is $$(9-2x)$$) we can find that maximum value of $$V$$ will be obtained when $$x=1.8$$

Therefore, Maximum Volume $$=110$$

This can be verified using the graph

• Note that $6.2$ and $1.8$ are approximations, not exact values. There is no need to substitute $x \approx 6.2$, because we know that $x \le \frac92$ given that one of the side lengths is $9$. Nov 3, 2022 at 16:10
• Also the statement "derivative equals zero means the the point is either a maximum or a minimum" is not true; consider, for example, $y=x^3$ at $x=0$. Nov 3, 2022 at 18:24
• He did not take 6.2 into calculation but did not mention why.. Nov 3, 2022 at 22:01
• @Théophile thanks for teaching me that!
– neo
Nov 4, 2022 at 5:49
• @neo Glad to help. If you want to learn about some much wilder behaviour related to points where the derivative is zero, check this out: en.wikipedia.org/wiki/Cantor_function. Nov 4, 2022 at 16:56

The extreme values of "nice" functions (such as polynomials, which is what $$V(x)$$ is) occur either at boundary values or at points where derivative is zero.

Here, first get the "boundaries" of $$x$$, for example, $$x$$ cannot be negative so lower boundary is zero; also it cannot exceed the box's dimension. Find out $$V$$ at such points.

Then, find out $$V$$ at points where $$V'(x)=0$$

The maximum of all these $$V$$s is the requisite value.

Continuing from neo's approximate root if we take $$x\approx 6.2$$ then length of one side of the box would be $$9-2\times 6.2 <0,~$$ which is discarded.

Next taking second root for $$4 x^2-32x+45 =0,~ x\approx 1.820551 ;$$

Volume $$= 1.820551 (9-2 \times 1.820551)(15-2 \times 1.820551)\approx 110.81908 ;$$

Sign of next (second) derivative $$8(x-4)$$ is negative hence volume is confirmed maximum.

• This is incorrect; see the comments on neo's answer. Nov 3, 2022 at 18:23