An open box is to be made from a square piece of material, 36 centimeters on a side by cutting equal squares with sides of length x from the corners and turning up the sides (see figure).

Question

(b) Determine the domain of the func V.

0 cm < x < 18 cm

0 cm < x < 36 cm  

  0 cm < x < 6 cm

x < 18 cm

x > 36 cm

(c) Use the table feature of a graphing utility to show various box heights x and the corresponding volumes V. (Fill in the table below.)

Box Height, xBox Volume, V
1 cm	cm3
2 cm	cm3
3 cm	cm3
4 cm	cm3
5 cm	cm3
6 cm	cm3
7 cm	cm3

Use the table to estimate a range of dimensions within which the maximum volume is produced.

The x-value is between 1 cm and 3 cm.

The x-value is between 3 cm and 5 cm. 

 The x-value is between 5 cm and 7 cm.

Use the graph and the range of dimensions from part (c) to find the x-value for which V(x)

 is maximum. (Round your answer to the nearest whole number.)

x =_____  cm

Answer 1

b) the domain of the function V is the input x, meaning the span of the side length of squares you can cut out of the original 36 inches-on-a-side material you are using.

you cannot cut out a negative side-length, so that tells you x cannot be negative. can x be zero? probably not, since the missing squares are what enable you to fold up the material to make the box.

what is the largest square you could cut out, regardless of sensibility? if you go as large as 18 on a side for your cut-out, then you have, in essence, cut out the entire original material.

so x has to be bigger than 0 and smaller than 18, which would be written 0 < x < 18.

c) you need to enter into your table function your volume formula, tailored to this problem. volume of a box = (length)(width)(height). in this case, length = width. since the original material is 36, and you are cutting out an x-by-x square from each corner, the width/length of the box will be 36 - 2x. the height of the box will be the x you chose for the squares you cut from each corner.

V = (36 - 2x)(36 - 2x)(x). you may need to accept "y" as a substitute for your desired V when you type it in the table function. technically, your function is V(x). here are values you get.

V(1) = 1156

V(2) = 2048

V(3) = 2700

V(4) = 3136

V(5) = 3380

V(6) = 3456

V(7) = 3388

even without graphing, you can see from the table that x = 6 is the whole number where the graph has a maximum, which fits in the span of 5 < x < 7.

i tried to embed a desmos graph, but wyzant will not let me.