YOU DO NOT HAVE TO DRAW THE DIAGRAMS JUST NEED ANSWERS FOR THE BLANKS! Consider

Question

YOU DO NOT HAVE TO DRAW THE DIAGRAMS JUST NEED ANSWERS FOR THE BLANKS!

Consider Consider the following problem: A box with an open top is to be constructed from a square piece of cardboard, 3 ft wide, by cutting out a square from each of the four corners and bending up the sides. Find the largest volume that such a box can have. a square (a) Draw several diagrams to illustrate the situation, some short boxes with large bases and some tall boxes with small bases. Find the volumes of several such boxes. (b) Draw a diagram illustrating the general situation. Let x denote the length of the side of the square being cut out. Let y denote the length of the base. (c) Write an expression for the volume V in terms of both x and y nding up largest (d) Use the given information to write an equation that relates the variables x and y. (e) Use part (d) to write the volume as a function of only x. V(x) = (f) Finish solving the problem by finding the largest volume that such a box can have. fi 3

Explanation / Answer

c)The length of the base of the box = 3 - 2X

d)where X = length of the square that needs to be cut at each corner.

The depth of your box is = X

hence your box volume will be

V = (3 - 2X)(3 - 2X)(X)
V = (9 - 12X + 4X^2)(X)

Expanding and rearranging terms,

V = 4X^3 - 12X^2 + 9X

e)Given expession for V, next step is to differentiate "V" with respect to "X"

(dV/dX) = 12X^2 - 24X + 9

To determine the value of X for maximum volume of the box, equate (dV/dX) to zero and solve for X.

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