Previous Problem ListNext (1 point) (8.7.7) Figure 1 Figure 2 A/(n) 3 in x 8 in

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Previous Problem ListNext (1 point) (8.7.7) Figure 1 Figure 2 A/(n) 3 in x 8 in piece of cardboard has squares cut out of each corner in order to make a box (see Figure 1) Let z represent the length of a cut-out square (the height of the box). tion () tor the volume of the box in termis of z (a) Find a function the volume of the box in terms of z (b) Find the feasible domain of the function when considering any physical restraints. Write your answer as a compound inequality involving z Domain of V(z) (c) Using the graph of V(z) shown in Figure 2, determine the dimensions that yield the maximum volume. Round your answers to the nearest tenth. Height in Width: in Length: in Help: Click here for heip entering formulas or click here for help entering inequalites It does not matter which side you choose to be the width or lenath of the box

Explanation / Answer

a).The dimensions of the piece of cardboard are 3 inches by 8 inches. After cutting off the corners, and on folding the sides, the dimensions of the base of the box are 3-2x inches by 8-2x inches. Then, the volume of the box is given by V(x) = length*width*height = (8-2x)(3-2x)x = (24-22x+4x2)x = 4x3-22x2+24x. Thus, V(x) = 4x3-22x2+24x.

(b). Since the shorter side of the piece of cardboard is 3 inches and since the square corners have side admeasuring x inches, we must have 2x < 3 or, x < 3/2. Thus the domain of V(x) is 0 < x < 3/2

.

(c ). As per the graph of V(x), there is a maximum at (0.7,7), i.e. when x = 0.7 inches, the volume V(x) is maximum. Thus, for maximum volume of the box,

height (x) = 0.7 inch, width = 3-2x = 3-1.4 = 1.6 inches and length = 8-2x = 8-1.4 = 6.6 inches.

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