Question

What is the maximum volume in cubic inches of an open box to be made from a 16-inch by 30-inch 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. (10 points

Answer 1

Maximum Volume = 725.93 in³ -

Explanation:

Given

Length, L = 16 inch

Width, W = 30 inch

Let x = squares of equal sides from the four corners

With that, Length and Width has been reduced by ½(4sides) = ½, of 4x = 2x

New Length = 16 - 2x

New Width = 30 - 2x

Calculating the volume

Volume, V = Length * Breadth * Height

V = (16 - 2x) * (30 - 2x) * x

V = (16 - 2x) * (30x - 2x²)

V = 480x - 32x² - 60x² + 4x³

V = 4x³ - 92x² + 480x

To obtain maximum value;

We calculate dV/dx = 0

dV/dx = 12x² - 184x + 480

12x² - 184x + 480 = 0

Solving for x

12(x² - 184x/12 + 20) = 0

12(x² - 46x/3 + 20) = 0

Factorize;

12(x² - 10x/3 - 12x + 20) = 0

12(x(x - 10/3) - 12(x - 10/3)) = 0

12(x-12)(x-10/3) = 0

At this point

12 ≠ 0

So, the critical points are;

x = 12 and x = 10/3

Putting these values of x in Length and Breadth

L = 16 - 2(12)

L = 16 - 24

L = -8.

But length can't be negative.

So, x = 12 is invalid

Our critical point is x = 10/3

To get the maximum volume

V = V = (16 - 2x) * (30 - 2x) * x becomes.

V = (16 - 2(10/3)) * (30 - 2(10/3)) * 10/3

V = 725.9259259259259

V = 725.93 in³ ---- Approximately