Question

You are planning to make an open rectangular box from aa 4141​-in.-by-8181​-in. piece of cardboard by cutting congruent squares from the corners and folding up the sides. What are the dimensions of the box of largest volume you can make this​ way, and what is its​ volume

Answer 1

Answer:

13058.83 cubic inches

Step-by-step explanation:

Given that a rectangular box is having dimensions as 41x81 inches.

Let x be the side of square cut from all the four corners.

The open box made would have height as x and length 41-2x with width 81-2x

Volume =

$V(x) = x(41-2x)(81-2x)\\V(x) =x(3321-244x+4x^2)\\V(x) = 3321x-244x^2+4x^3\\V'(x) = 3321-488x+12x^2\\V"(x) = -488+24x$

Equate first derivative to 0

We get applicable root as x = 8.642

Max volume = 13058.83 cubic inches