On a rectangular piece of cardboard with perimeter 11 inches, three parallel and equally spaced creases are made. The cardboard is then folded along the creases to make a rectangular box with open ends. Letting x represent the distance (in inches) between the creases, use a graphing calculator to find the value of that maximizes the volume enclosed by this box. Then give the maximum volume. Round your responses to two decimal places.

values of x that maximizes volume = in
Maximum volume= in^3

I'm so lost and don't know where to start any help would be greatly appreciated

To find the value of x that maximizes the volume of the box, we can use calculus and the concept of optimization. Here's a step-by-step guide on how to approach this problem:

Step 1: Understand the Problem
Read the problem carefully and identify the given information. We know that the perimeter of the cardboard is 11 inches, and there are three parallel and equally spaced creases.

Step 2: Define Variables
Let x represent the distance (in inches) between the creases.

Step 3: Formulate Equations
To maximize the volume of the box, we need to come up with an equation that represents the volume. Since the box is formed by folding the cardboard, the length of the box will be x inches, the width will be x inches, and the height will be the remaining part of the cardboard.

We know that the perimeter of a rectangle is given by P = 2(length + width). In this case, the perimeter is 11 inches, so we can write:
11 = 2(length + 2x)
Simplifying this equation, we get:
length + 2x = 11/2
length = 11/2 - 2x

The volume of a rectangular box is given by V = length * width * height. Substituting the values we have, we get:
V = (11/2 - 2x) * x * height

Step 4: Express Volume in Terms of x Only
To eliminate the height variable, we need to express it in terms of x. Since there are three equal creases, the distance between the first crease and the end of the cardboard is x/2. Therefore, the height is (11 - 5x)/2.
Substituting this into the volume equation, we get:
V = x * (11/2 - 2x) * (11 - 5x)/2

Step 5: Graph the Function
Using a graphing calculator, plot the graph of V = x * (11/2 - 2x) * (11 - 5x)/2. Make sure to set an appropriate window for the x-values.

Step 6: Find Maximum Value and Corresponding x
The maximum volume is the highest point on the graph. Use the graphing calculator to find the x-value that corresponds to this maximum point.

Step 7: Calculate Maximum Volume
Once you have the value of x that maximizes the volume, substitute it back into the volume equation to calculate the maximum volume.

Remember to round your final answers to two decimal places, as stated in the problem.

I hope this step-by-step guide helps you solve the problem!

This was answered by DrRuss on 1/21 at 6:39 am. I can't link so I pasted his answer below.

I would start with a drawing as I do for most problems. Draw a reactangle 4x one side and b the other.

The perimeter is then 4x+b+4x+b=11

8x+2b=11

if this is folded to a tube then the volume of the tube is bx^2, i.e. a tube with cross sectional area x^2 and length b.

so V=bx^2

rearrangen and substitute for b into the equation above gives

8x+2V/(x^2) = 11

or

8x^3+2V=11x^2

or

V=5.5x^2-4x^3

which you can plot to find max V

I got 1.54 in^3 as the max volume

but check the maths!

Ok but I still don't know how to get the values of x that maximizes volume = in

sorry, I didn't know that.

you should have said that in your 1st post above. the tutors would see that this question was answered and refer you to the original answer.