Please help! I do not know where to start. I'm so confused.

Question

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

(a) Write a function
V(x)
that represents the volume of the box.
V(x) =

(b) Determine the domain of the function.
0 < x < 36
0 < x < 18
x > 18
x < 18
0 < x < 9

(c) Use a graphing utility to create a table that shows box heights x and the corresponding volumes V.
Box Height x    Box Width     Box Volume V
1

2

3

4

5

6

7


Use the table to estimate the dimensions that will produce a maximum volume. In inches
length

width

height


(d) Use a graphing utility to graph V and use the graph to estimate the value of x for which
V(x)
is maximum.


The maximum point on the graph occurs at x =


Compare your result with that of part (c).
This _____ the maximum point found in part (c).

(agrees, does not agree)

Answer 1

Hi Blake,

 

We'll take each part of the question step-by-step and all will be clear:

 

 

Part A (Volume Function):

 

    1.  Let's begin by seeing that each side of the square will be reduced          by 2x inches (x inches at each corner of a side).

 

    2.  The height of the box will be length of a turned-up corner

         (x inches).

 

    3.  Thus, we have the dimensions of the box:

             (a)  length = (36 - 2x) inches.

             (b)  width = (36 - 2x) inches.

             (c)  height = x inches.

 

    4.  We can now write the volume function of the box in terms of x:

             Volume = length x width x height

             V(x) = (36-2x)(36-2x)(x)

 

 

Part B (Domain of the Function):

 

    1.  The domain of the function will be all input x values which give a

         legitimate output y value.

 

    2.  For our function, we cannot have a volume of zero and we cannot

         have a negative volume (any value of x which would cause the y

         value to be less than zero). 

 

    3.  If x = 0 then the volume will be zero, since x is one the three

         factors of the function.  Also if the (36 - 2x) factors evaluate to

         zero then the volume will also be zero:

             36 - 2x = 0

             2x = 36

             x = 36/2

             x = 18 (will cause volume to be zero) 

 

    4.  If x is less than zero then volume (y) will also be less than zero,

         since one of the three factors of the function will be negative.

 

    5.  Putting the criteria from steps 3 and 4, we can write the domain

         of the function:  0 < x < 18

 

 

Part C (Graph of Function):

 

Below is the link to our graph of the Volume Function:

 

https://dl.dropbox.com/s/55y9bn6uz9dgc08/Graph_of_Volume_Function.png?raw=1

 

The graph was generated on the Casio Prizm graphing calculator, but could also have been easily generated on the TI 8x series of graphing calculators used by most schools.

 

The points of interest on the graph are:

 

  (a)  Maximum box volume is 3,456 in³  for an input height of 6

        inches.

 

  (b)  An input of height of 18 inches will cause the box volume to go

        to zero (our maximum domain value in answer to Part B).

 

Our table of values used for plotting our graph

(h = height, v = volume):

 

    1.  (h,v) = (1,1156)

    2.  (h,v) = 3,2700)

    3.  (h,v) = (6,3456) 

    4.  (h,v) = (9,2916)

    5.  (h,v) = (12.1728)

    6.  (h,v) = (15,540)

    7.  (h,v) = (18,0)

 

Trust all the above is helpful to your understanding of how we were able to write and graph the Volume Function.

 

Thanks for submitting this problem and glad to help.

 

God bless, Jordan.