# Maximum volume of a box with corners cut out calculator

By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. How do I approach this question when there are the corners cut out? I understand I need to label important things with variables and find an appropriate formula, however what do I do when it comes to the corners?

When you remove the four corners of the cardboard, you obtain exactly the unfolded box.

### Maximum Volume of a Cut Off Box

The base of the box is the rectangle defined by the four inner vertices. The rest of the carboard are the folded front, back, left and right sides. If you fold up the sides the box is the open parallelepiped sketched on the right. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. How do I find the maximum volume for a box when the corners are cut out? Ask Question. Asked 4 years, 4 months ago.

Active 4 years, 3 months ago. Viewed 3k times. Alex M. Dec 14 '15 at Active Oldest Votes. Koncopd Koncopd 2 2 silver badges 9 9 bronze badges.

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Visit chat. Related 1.Ask Dr. Math Maximize Volume of a Box Dr. Math archives. Maximum Volume of a Box A rectangular sheet of cardboard measures 16cm by 6cm. Equal squares are cut out of each corner and the sides are turned up to form an open rectangular box.

## Maximum Volume of Lidless Box

What is the maximum volume of the box? Maximizing the Volume of a Box What dimensions should a rectangular piece of paper have to maximize the volume of the box made by cutting the corners out and folding?

Maximizing the Volume of a Box: Find Size of Square Cutout If I have a rectangular card of size AxB, how can I find the size of the square cutout that maximizes the volume of the box produced when the edges are folded up?

Open Box Problem Find the formula for the greatest volume box you can make from a sheet of cardboard with different-sized corners cut out of it. Maximizing the Volume of a Box I have a piece of glass that is 14" by 72". What dimensions would I need to make a glass cage with maximum volume? Maximizing Volume of a Cereal Box Why are cereal boxes the size they are?

Is it just to maximize volume? Maximum Volume: Making a Box from a Sheet of Paper I am making a box out of a sheet of paper by cutting squares out of the corners. Dimensions of a Cardboard Box A box with a square bottom and a volume of centimeters can be made by cutting 5-centimeter squares from the corners of a piece of cardboard Math Maximize Volume of a Box. Maximizing the Volume of a Boxa selection of answers from the Dr.One of the key applications of finding global extrema is in optimizing some quantity, either minimizing or maximizing it.

For example, suppose you wanted to make an open-topped box out of a flat piece of cardboard that is 25" long by 20" wide. You cut a square out of each corner, all the same size, then fold up the flaps to form the box, as illustrated below. Suppose you want to find out how big to make the cut-out squares in order to maximize the volume of the box. This applet will illustrate the box and how to think about this problem using calculus.

The applet shows the flat piece of cardboard in the upper left, and a 3D perspective view of the folded box on the lower left. Move the x slider to adjust the size of the corner cutouts and notice what happens to the box. When x is small, the box is flat and shallow and has little volume.

When x is large, the box it tall and skinny, and also has little volume.

Somewhere in between is a box with the maximum amount of volume. Obviously, the smallest x can be is zero, which corresponds to not cutting out anything at all. What is the largest possible value for xand why? The volume of the box, since it is just a rectangular prism, is length times width times height. The height is just the size of the corner cut out x in this problem.

The length and width of the bottom of the box are both smaller than the cardboard because of the cut out corners. The graph of this function is shown in the upper right corner. As you move the x slider, the corresponding point moves along the graph, and the volume for that particular x value is also shown in the upper corner of the graph.

Prior to calculus, you might have solved this problem by graphing it on a calculator and finding the highest point on the graph. But, you can do better by finding the derivative of the volume function, setting this equal to zero and solving to find the critical points, determining which is a local maximum, and lastly comparing the volume at this point with the volume at the endpoints which we don't really need to do in this problem, since the volume is zero at the two ends of the relevant domain for x.

Volume of Open Box Made From Rectangle with Squares Cut Out

It is easier for most people to find the derivative by first exanding the volume formula into and then finding the derivative, which is Setting this equal to zero and solving e. The first of these is outside the allowable values for xso the solution is the second.

### Optimization: Maximizing volume

In the applet, the derivative is graphed in the lower right graph. Note that the derivative crosses the x axis at this value, and goes from positive to negative, indicating that this critical point is a local maximum. At the bottom of the applet are input fields for the length and width of the cardboard. Play around with different values to see how it affects the solution and the shape of the volume function. Note that this applet automatically computes the limits for the graphs i.

The applet also displays the formula for the volume in terms of xLand W as well as the formula for the derivative, but it computes the derivative without expanding i. Home Contact About Subject Index. This device cannot display Java animations.One of probably most regular problems in a beginning calculus class is this: given a rectangular piece of carton.

If by cuts parallel to the sides of the rectangle equal squares are removed from each corner, and the remaining shape is folded into a box, how big the volume of the box can be made?

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A rare calculus source would miss an opportunity to discuss such a practical application of differential calculus. A temptation is also great to use modern technology, say, graphic calculators or dynamic geometry software, like Geometer's Sketchpad, to solve the problem graphically in an attempt to reduce the proficiency required of students to understand the problem and approach its solution. Indeed, one such attempt intended for the middle school has been made at the On-Math, the Online Journal of School Mathematics.

The article, however, as the whole magazine, is only available by subscription to the NCTM members. Which supplied me with a motivation to create a publicly available online discussion. Let a and b denote the horizontal and vertical dimensions of the rectangle. At the outset, these may be reasonably designated as its length and width. However, technology permits modifications: there are three points that control the dimensions of the rectangle and the squares.

In particular, it is possible to make the horizontal dimension of the rectangle smaller than its vertical dimension, which would beg for a reverse appellation: width horizontal and length vertical. This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. In any event, assume that x is the common side of the squares. The potential base of the box under construction has dimensions: horizontal a - 2x and vertical b - 2x.

After the remaining four rectangles are folded to form a box, x also becomes the height of the construction. The volume of the box is thus given as the function of x by. The graphs of such polynomials all look about the same.

So that the two non-zero roots are positive. Following the graph from left to right, it crosses diagonally the third quadrant and meets the horizontal axis at the smallest root 0, in our case. It continues to grow up to a local maximum, after which it turns downwards and crosses the axis second time, later turning again upwards at the local minimum, and continues from there growing without bound, on the way crossing the axis the third time. The two positive roots of the polynomial are then equal, and the horizontal axis is tangent to the graph at the double root.

We are thus interested in the values of x that satisfy. We expect the equation to have two roots: one corresponding to the local maximum and the other to the local minimum of v x. Of interest to us is the smallest of the two. We can apply the quadratic formula to find both.

The smallest of the two is obtained by selecting the minus sign. It's important to understand that technology here provides only an approximate solution to the problem.One of the most practical uses of differentiation is finding the maximum or minimum value of a real-world function. In the following example, you calculate the maximum volume of a box that has no top and that is to be manufactured from a inch-byinch piece of cardboard by cutting and folding it as shown in the figure.

What dimensions produce a box that has the maximum volume? If a manufacturer can sell bigger boxes for more money, and he or she is making a million boxes, you better believe he or she will want the exact answer to this question:.

Express the thing you want maximized, the volume, as a function of the unknown, the height of the box which is the same as the length of the cut.

Find the critical numbers of V h in the open interval 0, 15 by setting its derivative equal to zero and solving. And because this derivative is defined for all input values, there are no additional critical numbers.

So, 5 is the only critical number. Thus, the dimensions of the desired box are 5 inches by 20 inches by 20 inches.Given a rectangular sheet of cardboard 15 in. If a small square of the same size is cut from each corner and each side folded up along the cuts to from a lidless box:.

## What size squares should be cut from each corner to obtain a maximum volume?

The first thing to realize from this figure is that the size of the square that is to be cut from each corner must be less than 7. This is true because if the size of the square was 7. There are a couple of different ways one can go about solving the two problems asked above. One could answer them graphically, with the help of a spreadsheet, using hand written mathematics and solving equations, etc.

The graphical and spreadsheet methods will be used here. In order to get a graphical representation of this problem we need an equation for the volume of the box. Now graphing this equation remembering that x must be less than 7. Figure 2 shows this graph. From this figure, notice that the maximum volume is a little over cubic inches and this occurs when the size of the square cut out of each corner is about 3 in. Now we have narrowed down the range for which the size of the square cut from each corner can be.

It is in the range between 2. This is seen in Figure 3. Now with the range narrowed down, we can go to a spreadsheet program and let the spreadsheet do the calculations for many different possible values of the size of the square to be cut out and determine the size of the square that when cut out creates a box a greatest volume. One way to do this is to put in one column values of the width of the square to be cut out, starting with 2. Then in another column determine the volume for each of these values.

Then look at the different volumes that were just calculated and find the values of the width of the square where the volume is going up, then levels off and then starts going back down. Now use the smallest of these three widths and create another column similar to the first column you created but you only need to let the widths of the square range between the the smallest and largest of the three widths you just determine to be the ones that zeroed in on the maximum volume.

With multiple repetitions of this one can find a value for the width of the square that will give the greatest volume.

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With each repetition, you gain one more decimal place of accuracy. Table 1 below illustrates the initial calculation of the volume of the box for values of the width of the square between 2.

Side of Square in.Math [ Privacy Policy ] [ Terms of Use ]. To do this, you have to cut out squares in the corners of the paper. When using a square, I noticed that the side of the square equals a sixth of the size of the length of the paper.

I only worked this out by noticing patterns in several tables that I drew up.

I asked a friend of mine and apparently it has something to do with differentiation. I don't know anything about this, so could you explain it in simple terms to me so that I can explain how to show that the length of the side of the square equals a sixth of the length of a side of the piece of paper?

I would be most grateful if you could help me with this. In your case we will say that the width of the piece of paper is W. I am also going to assume that we want an open box no lid.

The value you want to find is x. This is the value we are looking for. But we don't want to graph the equation and look for the maximum. Maximums and minimums are found when the slope is zero. When the slope is zero the graph is changing direction to go back the way it came and that is a maximum or minimum.

If we differentiate this equation we get an equation of the graph's slope. And wherever that equation is equal to zero is a maximum or minimum. The differential of V is the slope of the graph of V. If you need a better explanation, please write back. Search the Dr. Math Library:.