Question

BOXES Simon will make a box without a top by cutting out corners of equal size from a 22 inch by 15 inch sheet of cardboard and folding up the sides. Which of the following is closest to the greatest possible volume of the box?

a. 330 in3
b. 144 in3
c. 432 in3
d. 1296 in3

Answer 1

Answer:

Volume of box is $432 in^3$

Step-by-step explanation:

Let x be the side of square corner

Length of box after cutting corners =22-2x

Breadth of box after cutting corners =15-2x

Height =x

Volume of box = (22-2x)(15-2x)x

Volume of box =$4x^3-74x^2+330x$

Differentiate w.r.t x

$\frac{dV}{dx}=12x^2-148x+330$

Substitute derivative =0

$12x^2-148x+330=0\\x=\frac{37-\sqrt{379}}{6},\frac{37+\sqrt{379}}{6}\\\frac{d^2V}{dx^2}=24x-148$

at $x=\frac{37-\sqrt{379}}{6}$

$\frac{d^2V}{dx^2}=24x-148=24(\frac{37-\sqrt{379}}{6} )-148=-77.87 < 0$ hence maximum

at  $x=\frac{37+\sqrt{379}}{6}$

$\frac{d^2V}{dx^2}=24x-148=24(\frac{37+\sqrt{379}}{6} )-148=77.87>0$ hence minimum

So,Volume of box =$4x^3-74x^2+330x=4(\frac{37-\sqrt{379}}{6} )^3-74(\frac{37-\sqrt{379}}{6} )^2+330(\frac{37-\sqrt{379}}{6} )=432.23 in^3$

Hence Volume of box is $432 in^3$