{Use of Tech} Optimal boxes Imagine a lidless...

Question

{Use of Tech} Optimal boxes Imagine a lidless box with height h and a square base whose sides have length x. The box must have a volume of 125 ft³.

b. Based on your graph in part (a), estimate the value of x that produces the box with a minimum surface area.

Accepted Answer

Below there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Consider a cuboidal 8 cubic meter container with a square base and no top. Each side of the base measures A, and the height of the container is B. Sketch the graph of the function that depicts the surface area of the container of A for A is greater than 0. Using the graph, estimate the value of a that minimizes the surface area and round your answer to two decimal places. Awesome. So it appears for this particular prompt we're asked to solve for two separate answers. Our first answer we're trying to solve for is we're trying to create a sketch of the graph of the function that depicts the surface area of the container. So we're trying to figure out a graph for this SFA, which is the function for the surface area of the container. That's our first answer. Our second answer is we're asked to use our graph to help us estimate the value of A, which minimizes the surface area, and we're also asked to take this value of A and round our answer to two decimal places. So that's our second and final answer. So we're trying to figure out for our second answer once again, what is the estimated value of A. So with that in mind, We need to first off, we need to once again note and recall that this particular cuboidal container with a has a square base, and we also need to recall and note that each side of the base is A as specified in the prom itself. So each side of the base will measure some value A, and the height of this container will be B. And we need to know once again it's very important that this particular container has no top, and we're also told that the volume of the container is 8 cubic meters, so let's make a note of that, that the volume V is equal to 8 cubic meters. Awesome. And what we're going to be doing, as we mentioned earlier, is we're going to be creating a sketch of the graph of this function for S of A, which is the surface area of the container, and we need to know once again that A is going to be a positive number because A is greater than 0. And when we take our graph, we're going to use our graph to help us find the value of A for which there for which there will be a minimum surface area. So with that in mind, let us focus now on trying to create an equation that helps us solve for the volume of this container, and for a moment here, let's ignore the units. So we know that our volume is going to be equal to 8, so our expression to help us solve for the volume is we need to take the base side A, so we need to take a base multiplied by a base. Once again, so the side of the base, so a side base multiplied by the side of the base, multiplied by the height of the container will be equal to 8 the volume. So when we simplify, we will get a squared multiplied by B is equal to 8. But we need to figure out what the value of B the height is, so if we divide both sides by a square to get B all by itself, we'll find that B. is going to be equal to 8 divided by a 2. Fantastic. So now, our next step is we need to take And now we need to take this information. We need to make sure we hold on to it because we're going to need it in a little bit. But our next step is we need to figure out an expression, or rather an equation for the total surface area of this container, excluding the top because once again there is no top to this container. So let's call the total surface area TS and Total surface area, once again, excluding the top, and we're, once again, we're trying to create an equation for the total surface area, not including the top, and this will be equal to A multiplied by A plus A multiplied by B plus A multiplied by B plus A multiplied by B plus A multiplied by B. Now as you can see, it's 4 times, because there's 4 sides, so +1234. So when we simplify, we will get that F of A, which is our function. So F of A is equal to A2 + 4 multiplied by AB. Awesome, or a multiplied by B. But to keep things simple, we could just say 4 multiplied by a B. So now we need to substitute in our value for B equals 8 divided by a squared. That's when our value for B comes in handy. So now we need to simplify our expression. So writing it out first, we need to say that F of A is equal to A2 + 4 multiplied by a multiplied by 8 divided by a squared. Fantastic. So now, like I mentioned before, we need to simplify. So when we simplify, we will find that F of A is equal to A2 plus. 32 divided by A. Fantastic. So now is the easy part. So now all we need to do is we need to take our expression for F of A, which I said F of A, but it's still the same as SA, the surface area, but you know, we wrote it as a function, potato potato, but you don't need to write SFA, but you can if you want to. But we need to take this expression, this function, which is F of A, which is our value for SFA. is equal to a 2 + 32 divided by A, and we need to plug this into some sort of graphing software that's easy to use that we understand how to use. And when we plug in, This function into our graphing software we should achieve the following graph. And as you can see looking at our graph, we have a shape or rather a curve, that's what we, the proper term, we have a curve in purple and it's parabolic looking in shape. And as we could see, in order to determine the value for A, so the value of A, once again that minimizes the surface area, so that means it will be the lowest point on this curve. So using my red pen here, the lowest point on our curve, estimating just visually on this graph. Which you, when you use your graphing software, there usually there's an option where you can like ask it to like point out what the lowest point on the curve is. And if you can like make sure you can indicate that in your graphing software, that is your value for A. So the lowest point on the graph, when we round to two decimal places, is going to be 2.52, so 2.52, that's our value for X. and our value for Y will be 19.049. So, the lowest point, once again on our curve is what we're going to be using to solve our value for A, and specifically our value for A corresponds to the value for the X coordinate. And once again, our Y coordinate is 19.049. So once again, the value of A for the minimum surface area will be approximately 2.52, and this is when we round to two, once again, two decimal places, and it's units of measurement once again will be meters. So it's going to be approximately equal to 2.52 m. And that is our final answer. Hooray, we did it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

{Use of Tech} Optimal boxes Imagine a lidless box with height h and a square base whose sides have length x. The box must have a volume of 125 ft³.

b. Based on your graph in part (a), estimate the value of x that produces the box with a minimum surface area.

Key Concepts

Volume of a Box

Surface Area of a Box

Optimization in Calculus

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