If you want to produce an open-top box from a 16 in by 22 in flat piece of cardboard, what is the maximum possible volume (in in3) of the box? round your answer to the nearest whole number and omit units.

480

Step-by-step explanation:

Since, for making a box from a cardboard,

We need to cut four congruent pieces from each corner of the cardboard,

Let x be the side of a piece ( in inches ),

Given,

The dimensions of the cardboard are 16 in by 22,

So, the dimension of the box would be (16-2x) in by (22-2x) in by x in,

Thus, the volume of the box,

Differentiating with respect to x,

Again differentiating with respect to x,

For maxima or minima,

V'(x) = 0

By the quadratic formula,

Since, for x = 9.62, V''(x) = positive,

While for x = 3.05, v''(x) = negative,

Hence, volume is maximum for x = 3.05,

And, maximum volume,

Answer & Step-by-step explanation:

Use the slope formula:

The slope is the change in the y-axis over the change in the x-axis, otherwise known as rise over run. Insert values:

Simplify. Two negatives make a positive:

Solve:

The slope is 0. Plot the two points on a graph and connect with a straight line: