The postal service will accept a box for shipment only if the sum of its length and girth (the distance around) does not exceed 108 inches. What dimensions will give a box with a square end the largest possible volume?
[Math] Volume Optimization
calculusoptimization
Best Answer
Let $x$ be the length of a side of the square end, and let $y$ be the length of the box. As you say in your comment, the volume is given by $V=x^2y$, and the postal service’s restriction limits $4x+y$ to a maximum of $108$. Clearly we should use the entire allowance, so we want $4x+y=108$. However, you’ve solved this incorrectly for $y$: $y=108-4x$, not $\frac{108}{4x}$. Substituting the correct expression for $y$ into the volume formula, we get $V=x^2(108-4x)=108x^2-4x^3$.
You want to find the value of $x$ that maximizes $V$, knowing that $V=108x^2-4x^3$. How can you use the first derivative $\frac{dV}{dx}$ to find this value of $x$? (I’ll stop here to give you a chance to think about it; if you get completely stuck, leave a comment.)