Volume of an open box
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So x has got to be between 0 and 10. You can figure out what the exact value is if you like. It would look something like that. My previous answer divided by 24 gives me 3. Then I have 20 times negative 2x, which is negative 40x. Well, we could type that back in into our original expression for volume to figure what that is. So this has got to be greater than 0.

Or you could say that 20 has to be greater than or equal to 2x, or you could say that 10 is going to be greater than or equal to x, which is another way of saying that x is going to be less than or equal to 10. Also, the terms length, width, and height are just words to help you remember the formula. Given: You have a cardboard sheet that is 58 inches long by 36 inches wide. And so now let me do 200-- I probably could have done this slightly more efficiently, but this is fine-- minus the square root of 40,000-- 1, 2, 3-- minus 4 times 12 times 600. So this is going to be 20 minus 2x. So it's 20 minus 2x. So let me write this down.

You can imagine this flap right over here, if I were to fold it up like that, it now would look like this. We'll make L equal to S and W equal to S since they are both equal to each other. The correct way write out volume is with your type of measurement cubed. What is the biggest that x height can be? It is computed as 37. What x values gets my derivative to be equal to 0? So this is, the x's that satisfy this is going to be x is going to be equal to-- So negative b.

To start viewing messages, select the forum that you want to visit from the selection below. And it looks like we did get the right range. You subtract this x, you subtract this x, and you get this distance right over here. We already know that this corner right over here, which is made up of when this side and this side connect when you fold these two flaps up, that's going to be the same height over there. So let me do that.

After exploring an applet available on the Internet, the students find x by creating a spreadsheet that evaluates and graphs the volume of the box as a polynomial function of x for many values of x. And I want to maximize it by picking my x appropriately. Well, first of all, you don't need to be given the height, width, volume, and length. That's a different shade of yellow. And so first let me set my range appropriately before I attempt to graph it. H is equal to the height. Let me make my best attempt to draw it.

An instructor version is available by request. But I'm just going to cut to the chase and try to use the quadratic formula here. Because the length and width equal 30 โ 2 h, a height of 5 inches gives a length and width of 30 โ 2 ยท 5, or 20 inches. So what would the volume be as a function of x? This means that if the measurements of the sides were in inches, then the answer is in inches cubed or inches 3. Problem: Congruent squares x by x square inches in size, red in color in the diagram above are cut from the corners of the cardboard. We learned earlier that the surface area of a flat rectangle was the length times the width, but that was just a flat two-dimensional object. So minus 4 times 12, 12 times 600-- I still didn't give myself enough space-- times 600.

How to Use Differentiation to Calculate the Maximum Volume of a Box One of the most practical uses of differentiation is finding the maximum or minimum value of a real-world function. So let me give myself some more space. In the previous example, this means we could fit eight separate 1 foot cubes in our box. And my maximum y-value, let's see what would be reasonable here. Which I'll divide by 24, and I get 12.

I just randomly picked that. In the next video, we'll try to solve it analytically using some of our calculus tools. So at least based on the level of zoom that I have my calculator right now, this is a pretty good approximation for the maximum value that my function actually takes on. You may have to before you can post: click the register link above to proceed. In the last video, we were able to get a pretty good sense about how large of an x we should cut out of each corner in order to maximize our volume. And now let me do the situation where I subtract what I had in the radical sign. The that is to be minimized is the surface area s while the volume v remains fixed at 108 cubic inches.

Look at the box drawing below: You see that when dealing with a box, or three dimensional object, we have to consider three measurements: length, width, and height. Volume is the measurement of how much space a three dimensional object takes up. My minimum y-value, this is essentially going to be my volume. Well, in order to do the second derivative test, you have to figure out the second derivative is. Then, determine depth of water in the basin at a point, and diviide this term by total possible depth at that po โฆ int to determine ratio of fullness. And before we do it analytically with a bit of calculus, let's do it graphically. And before I even do that, it'll simplify things so I don't have to use the product rule in some way and then have to simplify that, let me just multiply this expression out.

This whole distance is 20. More Geometry Subjects Back to Back to. We want to make a list of pairs of values for height and volume, so we will use List on the calculator. What is our volume when x is equal to 10? Measure the height of the box. If you're given any three of those numbers, you can calculate the fourth one.