# An open-top box is to be made as follows...

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An open-top box is to be made as follows: squares of a certain size will be cut away from each of the four corners of a 20" x 30" rectangle, and the ends will be folded upward to form the corner seams, as shown. How big should the square cutouts be in order to maximize the volume of the resulting box?

#2:

Let f(x) = ((x-2)2) / (1+x)

a. Use derivatives to identify all local maxima and/or minima of f, if any.

b. Use calculus to find the highest and lowest values attained by the function f on

the interval 0 (greater than or equal to) x (less than or equal to) 5.

#3:

Let r(x) = (2/3)x3 - x - (1/3)x4 + (1/20)x5 + 1 :

Find the second derivative and set it equal to zero, in order to find inflection points.

#4:

Sketch continuous functions g and h satisfying the given conditions. Please sketch carefully enough to show the concavity clearly, and label axes and tickmarks where appropriate:

h ' (x) is negative for x < 1 and h' (x) is positive for x > 1; h" (x) is positive for

0 < x < 2, and h" (x) < 0 elsewhere

#### Solution Preview

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An open-top box is to be made as follows: squares of a certain size will be cut away from each of the four corners of a 20" x 30" rectangle, and the ends will be folded upward to form the corner seams, as shown. How big should the square cutouts be in order to maximize the volume of the resulting box?

Answer:

The height of the box is

The width of the box is 20-2x

The length of the box is 30-2x

The volume of the box is

In order to find the max volume, the first derivative will be taken

When , the x and V values could be max.

is discarded since 10" max can be cut out.

So will ...

#### Solution Summary

This solution is comprised of a detailed explanation to use derivatives to identify all local maxima and/or minima of f, if any.