construct box from 8.5 x 11 sheet by cutting sqrs at corners

[mop969]

You have an 8.5 inch by 11 inch piece of paper. You need to cut out four squares in each corner of the box so you can fold the sides of the paper and create a volume for the box.

1. Write an equation that represents the volume of the box.

2. What is the domain of the volume equation for the box.

3.What is the range of the volume equation for the box.

4.What are the x intercepts of the volume equation.

5.How do you graph the volume equation and what should it look like.

6.What does the restricted domain mean and how do you find the restricted domain. Also, what does the restricted domain of the graph mean.

7.What does the restricted range mean and how do you find the restricted range. Also, what does the restricted range of the graph mean.

8.What is the dimensions of the 4 cut squares need to be to have the maximum volume for the box. Explain your results. Also, will these previous steps work for all type of material such as sheet metal and aluminum.

 

[Deleted member 4993]

Re: The Box Maath Problem

You have an 8.5 inch by 11 inch piece of paper. You need to cut out four squares in each corner of the box so you can fold the sides of the paper and create a volume for the box.

1. Write an equation that represents the volume of the box.

2. What is the domain of the volume equation for the box.

3.What is the range of the volume equation for the box.

4.What are the x intercepts of the volume equation.

5.How do you graph the volume equation and what should it look like.

6.What does the restricted domain mean and how do you find the restricted domain. Also, what does the restricted domain of the graph mean.

7.What does the restricted range mean and how do you find the restricted range. Also, what does the restricted range of the graph mean.

8.What is the dimensions of the 4 cut squares need to be to have the maximum volume for the box. Explain your results. Also, will these previous steps work for all type of material such as sheet metal and aluminum.

Please share with us your work/thoughts - indicating exactly where you are stuck - so that we know where to begin to help you.

 

[mop969]

Re: The Box Maath Problem

I understand that the volume equation for the box is based on (Length)(Width)(height) which equals (8.5 - 2x)(11 - 2x)(x). The rest I am lost thanks for help.

 

[galactus]

Re: The Box Maath Problem

Here is an example problem. Use this template to answer your questions. OK?.

Let's say the box has dimensions 16 by 30 and we have to cut squares of equal size in the corners.

Let x=the length of the sides of the squares.

Let V= the volume of the box.

The volume is then given as \(\displaystyle V=(16-2x)(30-2x)x=4x^{3}-92x^{2}+480x\)

We can not cut out squares whose sides are more than 8. Can you see why?. That is the domain.

\(\displaystyle 0\leq x\leq 8\)

So, we know the solution is in the interval [0,8]. What is your domain?.

Differentiate the above function:

\(\displaystyle \frac{dV}{dx}=4(3x^{2}-46x+120)\)

Set to 0 and solve for x gives us \(\displaystyle x=\frac{10}{3}, \;\ x=12\)

12 is obviously out of our domain, so 10/3 is the solution we require.

Also, try our endpoints. They are 0 and 8. By trying 0,8,10/3 we see that 10/3 is the value of x that gives max volume.

That volume is 726 cubic units.

 

[blackie]

Re: The Box Maath Problem

Hi, I am also working on the same problem and I was wondering why and how do you Differentiate the function and how do you find the range and the restricted domain and range of the problem. Also, I am lost on how to answer question number eight of mop969 problem.

 

[stapel]

Hi, I am also working on the same problem and I was wondering why and how do you Differentiate the function...

The "how" should have been covered during the first third to half of the course, when limits and differentiation formulas were covered. The "how" relates to the First and Second Derivative Tests and finding max / min points of functions. Unfortunately, it is not reasonably feasible to attempt to replicate the weeks or months of classroom instruction or the hundreds of pages in your text. Sorry!

...and how do you find the range and the restricted domain and range of the problem.

Functions and their domains and ranges should have been covered back in one of your algebra courses. :shock:

Also, I am lost on how to answer question number eight of mop969 problem.

Since the complete explanation and worked solution has left you still perplexed, this, along with your lack of knowledge of the necessary background material, would seem to indicate a need for much more intensive assistance than can here be provided. Please seriously consider hiring a qualified local tutor and setting aside an hour or two a day for concentrated study sessions. :idea:

By working with you face-to-face, your tutor should be able to figure out many of the gaps in your knowledge, and back up and re-teach those topics. With hard work and a little luck, you may be able to get caught up in just a few weeks!

Eliz.

 

[blackie]

Re: The Box Maath Problem

I remeber how to do the derivitive but what is the restricted domain and range and I now understand the domain but how do you find the range is it by looking at the y partners when you graph the function. and I need explained the solution for number eight of the problem.

 

[Deleted member 4993]

Re: The Box Maath Problem

I remeber how to do the derivitive but what is the restricted domain and range and I now understand the domain but how do you find the range is it by looking at the y partners when you graph the function. and I need explained the solution for number eight of the problem.

What does your text-book say?

What does your class-notes say?

What does google say?

If your still stuck - write back telling us what you found - and exactly what part you don't understand.

 

[blackie]

Re: The Box Maath Problem

Here is what I believe to be right. The domain and range of the equation are all real numbers.And the inequalities represented the restricted domain and range. I do not understand what the x intercepts mean in the equation and do you find the maximum volume by plugging in all the x intercepts to find the highest value of the function.

 

[Deleted member 4993]

Re: The Box Maath Problem

Here is what I believe to be right. The domain and range of the equation are all real numbers.And the inequalities represented the restricted domain and range. I do not understand what the x intercepts mean in the equation and do you find the maximum volume by plugging in all the x intercepts to find the highest value of the function.

We plugged in x-intercepts of \(\displaystyle \frac{dV}{dx}(x)\) - to find the maximum of \(\displaystyle V(x)\) [ look up in your textbook - condition for maxima/minima of a function].

You are talking about two different equations - however one [\(\displaystyle \frac{dV}{dx}(x)\)] was derived from other \(\displaystyle \text V(x)\)

 

[blackie]

Re: The Box Maath Problem

So am I right about the domain and range and are you saying that to find the maximum volume I plug the x intercepts into the derivitive equation or the original equation. I believe it is thee original equation but I want to check.

 

[galactus]

Re: The Box Maath Problem

Yes, plug your maximizing value into the original function to find the max volume achieved. The derivative is the tool we use to find that value.

The volume equation is what we need to actually find the volume. Make sense?.


The width of your cardboard is 8.5 inches. We can only fold up so much to be practical. The max we can fold it is in half. That is 4.25 inches.

The domain is \(\displaystyle 0\leq x\leq 4.25\)

See what I mean?.

Domain is the region we're allowed to work in. When you put something in, what comes out is the range.

Here is an example:

Suppose we have \(\displaystyle y=\frac{1}{\sqrt{x-2}}\)

What is the domain?. Well, it is what we are allowed to use. In this case, no divsion by 0 is allowed nor any negatives in radicals because that results in a complex number. So, in this case, the domain would be everything but 2(gives division by 0) and any number less than 2(because that would result in a negative in the radical). So, the domain is \(\displaystyle (2,{\infty})\)

Now, the range is what we get when we plug in those values. The range would be \(\displaystyle (0,{\infty})\)

Suppose we plugged in x=2.0000000000000001, we get something huge approaching infinity. If we plug in 1000000000000, we get something approaching 0.

See?.

 

[blackie]

Re: The Box Maath Problem

So you mean that for the equation of the box the domain and range will be all real numbers but when the equation is applied to the problem which is the restricted domain the 0 to 5 value you said is used. But do you have to plug in all the x intercept values to find the volume or can you find it another way.