Question

An open box with a square base is to be made from a square piece of cardboard 24 inches on a side by cutting out a square from each corner and turning up the sides. If the volume V of the box is a function of the length x of the side of the square cut from each corner, for what value of x is V the largest

Answer 1

Answer:

x=4 Inch

Step-by-step explanation:

Length of the Square = 24 Inches

If a Square of Length x cm is cut out from each corner

Length of the Box = 24-x-x=(24-2x) Inches

Width of the Box =24-x-x=(24-2x) Inches

Height of the box = x inches

Volume of a Cuboid = Length X Width X Height

V(x)= x(24-2x)(24-2x)

Simplifying

V(x)=4x(12-x)(12-x)

To determine the value of x at which V is largest, we take the derivative of V(x) and solve for the critical points.

V(x)=4x(12-x)(12-x)

$V^{'}(x)=12(x-12)(x-4)$

Set the derivative equal to zero to obtain the critical points

$V^{'}(x)=12(x-12)(x-4)=0\\Since \: 12\neq 0\\(x-12)(x-4)=0\\x-12=0 \:or\: x-4=0\\x=12 \:or\: x=4$

x cannot be equal to 12 as it divides the length of the square cardboard into exactly two equal parts.

When x=4

V(4)=4*4(12-4)(12-4)=16*8*8=1024 Cubic Inches

When x=4 Inch, the volume, V of the open box is largest.