What is the maximum volume in cubic inches of an open box to be made from a 16 in by 30 in piece of cardboard by cutting out squares of equal sides from the four corners and bending up the sides? Your work must include a statement of the function and its derivative. Give one decimal place in your final answer.

Answer 1

length = 30 - 2x
width = 16 - 2x

V = (30-2x)(16-2x) = 4 (15-x)(8-x)
dV/4dx = (15-x)(-1) + (8-x)(-1)
= 2x^2 -23
fo min or max dV/dx = 0
x^2 = 23/2
x = sqrt(23/2)

Answer 2

did bottom area, not volume !
V = 4 x (15-x)(8-x)
V/4 = x(120 - 23 x + x^2)
V/4 = 120 x -23 x^2 + x^3
dV/4dx = 120 -46 x + 3 x^2 = 0 for min or max
solve quadratic
https://www.mathsisfun.com/quadratic-equation-solver.html

Answer 3

I get x = 12 or x = 3.3333333.... or 10/3
12 does not work. 16 - 2*12 is a very small side
So try 10/3
V = (10/3)(16 - 10/3)(30 - 10/3)