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.A box is formed by cutting squares from the four corners of a 11"-wide by 13"-long sheet of paper and folding up the sides. Let x represent the length of the side of the square cutout in inchesand let V represent the volume of the box in cubic inches. Write a formula that expresses V in terms of x.
Suppose the function f determines the volume of the box in cubic inches given a cutout length in inches. Write a function formula for f. What is the domain of f? Enter your answer as an interval. What is the range of f? Hint Tip: Use a graphing tool like www. Given that a box is formed by cutting squares from the four corners. If x is the length of side cut out in inches and V represents the volume of the box.
We find the range by finding the value of x such that the volume is maximum. Expanding the equation and simplifying gives. Questions are typically answered within 1 hour. Q: Symbolic notation for equations developed over a long period. Before symbols became commonplace, peo Q: Solve. Don't forget to Check! A: Refer to the questionwe have provide with centre as 2,4 and he tangent y —aixs. Q: Write an equation in slope-intercept form and in standard form for the line passing through 1,7 an A: To determine the equation of the line in the required format.
Q: One angle of a triangle is two times as large as another. The measure of the third angle is 20 degre A: Let x be the smallest angle of the triangle. The domain of f o g. Subscribe Sign in. Operations Management.Use this box volume calculator to easily calculate the volume of a rectangular box or tank from its length, width and height depth in any metric: mm, cm, meters, km, inches, feet, yards, miles The volume of a rectangular box can be calculated if you know its three dimensions: width, length and height.
Illustration below:. Measuring the sides of a rectangular box or tank is easy. The result from the calculation, using our volume of a rectangular box calculator or otherwise, will always be in the length unit used, cubed. Make sure to have all measurements in the same unit, if you are doing the math by hand, and if using our calculator - make sure to select the appropriate metric.
If you measured the lengths in inches, the result will be in cubic inches. If the length was in feet, the result will be in cubic feet, and so on for yards 3miles 3mm 3cm 3meters 3. To calculate the volume of a rectangular box or tank, you need to take three measurements, then multiply them.
In practical situations, you might have a plan or engineering schematic in which the measurements are all given making your task significantly easier.
That is fairly easy to do in your mind if the numbers are small, but quickly gets inconvenient if the numbers become large, where a volume of a box calculator such as the above gets really useful. It should be noted that "volume of a box" may not be a precise term, since if we take the box to be an abstract mathematical concept, its walls have zero thickness and the box itself has zero volume.
However, in practical situations we usually talk about the volume in terms of the space enclosed by the box or tank, and that is what the calculator above calculates.
If that is what you are calculating and want to know what it can hold, make sure to measure its internal, not external dimensions, otherwise you will need to then subtract the volume of the walls. Often one needs to calculate the volume of shipping containers, boxes, crates, tanks, and other transportation devices. Usually there will be a technical specification by the manufacturer, or at least you would know the standard based on which the container was produced, giving you minimum allowable dimensions.
If you want a more specialized calculator, which will tell you how many items if given dimensions you can fit, try our container calculator and our pallet calculator. The rectangular box is one of the most widely used body shapes across science, engineering, architecture, due to its good structural properties and resistance to force from different sides.
Most homes and rooms in them are rectangular boxes, most transportation containers are boxes or have a box-like shape. Calculating aquarium volume, swimming pool volume, or the volume of soil, mulch or plant nutrients you need for a given garden area are other common uses. Boxes and tanks from wood, steel, bricks, concrete, glass, and other materials are also easy to produce, compared to more complicated shapes. Another benefit of the rectangular shape is how easy smaller shapes can fit into larger ones, without the need of complex calculations, rotations and overall arrangement.
So, this box volume calculator has a lot of possible applications. If you'd like to cite this online calculator resource and information as provided on the page, you can use the following citation: Georgiev G.
Calculators Converters Randomizers Articles Search. Box length. Box width. Box height. Calculation results Box volume 60, cm 3. Share calculator:. Embed this tool! Volume of a box formula The volume of a rectangular box can be calculated if you know its three dimensions: width, length and height.Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.
We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page. Related Topics: More Lessons for Grade 9 Math Worksheets Examples, solutions, videos, worksheets, games, and activities to help Algebra students learn how to solve word problems that involve the volume of an open box.
How to find the dimensions of a cardboard used to form a box given the volume of the box? Example: The length of a piece of cardboard is five inches more than the width. The four corners are cut off so that the sides can be folded up to form a box. Each of the cut corners measures one and a half inches by one and a half inches, and the volume of the box is cubic inches.
What are the original dimensions of the piece of cardboard? Show Step-by-step Solutions. You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.I am horrible at math and need to do an extra credit project for my college algebra class.
I am given the volume of a cardbox box that I make. I start with a 8x10 piece of rectangular cardboard and cut squares out of the four corners to obtain a box with a volume of 40 cubic inches when the flaps are folded up. Thanks so much for helping! So I'm still math illiterate but I'm making my way. Is the x the size of the dimension I have to cut out to make it 40cubic inches? Bear with me. If you cut a square of length x out of each corner then you've changed both the length and width of the paper.
It's now length-2x and width-2x remember you cut two corners on each side. So volume would be x x x the sides only fold up once. You can do that with a y2 equation or just scroll through the graph until you find that point. So i need to figure out how big the four corners need to be that I cut out to obtain the volume. Update: Thanks so much for helping!
Answer Save. Volume Of A Box Calculator. Still have questions? Get your answers by asking now.System Simulation and Analysis. Plant Modeling for Control Design.
High Performance Computing. Maximum Volume of a Box This application is one of a collection of examples teaching Calculus with Maple. Steps are given at every stage of the solution, and many are illustrated using short video clips.
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Volume Word Problems
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. I need to find the dimensions of the box with maximum volume with faces parallel to the coordinate planes that can be inscribed in ellipsoid.
The corners go to. Eight times this maximum value gives the maximum volume, as per the hint. Sign up to join this community. The best answers are voted up and rise to the top.Volume of Open Box Made From Rectangle with Squares Cut Out
Home Questions Tags Users Unanswered. Maximize volume of box in ellipsoid Ask Question. Asked 8 years ago. Active 4 years ago. Viewed 8k times.
I seems to be missing something? Martin Sleziak Jiew Meng Jiew Meng 4, 15 15 gold badges 57 57 silver badges 89 89 bronze badges.
Active Oldest Votes. TonyK TonyK I don't think you can take this step for granted.
How to Use Differentiation to Calculate the Maximum Volume of a Box
And quite elegantly too. However, I still think you're missing a step.
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How to find the volume of a box using a graphing calculator?