Precalculus Textbook 8th Edition Section 1.7 Exercise 33 Solution

Precalculus Textbook 8th Edition Section 1.7 Exercise 33 Solution

Question:

A square of side x inches is cut out of each corner of a 10 in. by 18 in. piece of cardboard and the sides are folded up to form an open-topped box.

(a) Write the volume V of the box as a function of x.

(b) Find the domain of your function, taking into account the restrictions that the model imposes in x.

(c) Use your graphing calculator to determine the dimensions of the cut-out squares that will produce the box of maximum volume.

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Solution:

Precalculus 8th Edition Section 1.7 Exercise 33 Online Tutors AMBIPi Tutors

Since the dimension of rectangular cardboard is 18 by 10,

If a square of x unit is cut from each of the vertex of the rectangular cardboard, then a rectangular box is made of the dimension l = (18 – 2x), b = (10 – 2x) and h = x.

Then, the volume of box is

V = lbh

= (18 – 2x)(10 – 2x)(x)

= 4x(x – 5)(x – 9)

= 4x(x2 – 14x + 45)

= 4x3 – 56x2 + 180x

(a) Function of Volume of the box is V = f(x) = 4x3 – 56x2 + 180x.

(b) Domain the function f(x) = 4x3 – 56x2 + 180x is x belongs to (0, 5).

(c) To get the maximum volume of the box, the value of x = 2.06

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