A rectangular storage container with an open top is to have a volume of 10m3

Problem 16 Medium Difficulty

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Video Transcript

Here we have a manufactured 1 to make a rectangular box with no top, and the volume is 10 m. Cubed. We'Re told that the material for the base costs 10 per meter squared and the side material for the sides, cost 6 d per meter squared and were given a little bit of dimension. We are told that, given a certain width, then the length of the box needs to be twice that width and we're not given specific information on the height, but we can figure it out based on the fact that the total volume has to be 10 meter cube. So, let's take a look at that first and then our main goal is to find what are the dimensions to save money right to get the minimum cost. So let's go ahead and work on the left here. Regarding the volume we know, a volume of a rectangle is length times the width times height point, and we are told that the length is twice the width and we don't know the height, but we do know that the volume is 10 meter cube. So if i solved for mesten would be 10 divided by all this, and that would be divided by 2, w squared and i can reduce dividing top and bottom by 2, so 5 over w squared. So now i know that the height is 5 over w. Squared all right, also, okay, so next, let's go ahead and let's trade, an equation for costs, and then we can take the derivative and work towards minimizing the costs. All right. Let'S do that. Okay, so we're going to create a function of costs. That is a function of w and basically the cost is 10 dollars times. The area of the base will 10 dollars per meter squared times the area, the base plus 6 dollars per meter squared times the area of the sides. Okay, so let's go ahead and find the area base in terms of w and i'm gonna leave out the units for now to keep it simple for solving and the end will make sure answers incorrect units. Okay, so we have 10 times the area of the base. The area of the base is just the length times the width so to w times w point. Then we have 6 dollars per meter squared times the sides. Well, we have 2 sides that are like this, so we're going to have twice height times, width, but the height is 5 over w squared times a width, and then we have 2 that are this dimension, so we have plus, actually i can fit it all in The same parenthesis: let's do it like this: okay plus 2, both sides- both sides are made up of 2 w times the height and the height is 5 over w square all right. So let's clean this up because we want to take the derivative and we might also do it when it's cleaner and not messy okay. So this first term is 20 w squared and let's see if i look inside the parentheses on the right 1 of the cancels for each part, then it looks like i have. This is 10 plus 20 point, so that makes 30 and 30106 is 180 point. So i get plus 180 and we still have that w on the bottom, all right, let's go ahead and get a common denominator. I'M gonna multiply left this part by w over w. That will give me 20 w cube plus 180 over w. Actually, i want to take the derivative first, so, even though this is true, we are not going to do that yet because i want to take the derivative and i can frial more easily take the derivative when it's not that big fraction. So, let's just do derivative now, okay, so to find the minimum cost. We want to find check our critical point and see what's going on with our function, so we can take the derivative of my cost function with respect to w and that will give us 4040 w and then the derivative of 180 over w is minus 180 over W square and in case you're not seeing it i'll, do it at front if i think of it as 180 times w to the minus 1 for the function, then it's derivative. The 180 goes along for the ride, but the derivative i do power rule. So i get minus 1 times a w and i subtract 1 from the exponent, so you can see that that becomes minus 180 over w square. Okay. So that's my cost function derivative. Now i'm going to go ahead and do a common denominator. So i'm going to multiply my left term top and bottom by w square. That will give me 40 w cubed minus 180 over w square. Now i want to set this equal to 0 and or d n, and i'm going to try to find all my critical points. Well, the bottom: if w equals 0, then the bottom goes to 0, but that also is not a practical box so that i that's kind of like an edge point, but not a practical box. I don't think the manufacture is going to go for that 1, but we get the 0 case when the top goes to 0. So let's solve that real quick, so we're going to set the top equal to 020 w cubed minus 180 equals 0. So if i solve for w cube, i can do that by adding 180 to both sides and then dividing by 40 can cancel the zeros cause divide, both top and bottom by 10, and then w is then. The square root of santo square root cube root of 18 over 4 and that, if i plug into calculator, is about 1.651 point okay, so we want to find out if this is a minerum max. And so what we can do is basically check our critical points and are end points and that's 1 way to find out if it's a man, another way is to basically do kind of a sign chart. So i think this time i'm going to do a sign chart, i need to make the little room, and so i'm going to do it here in this little space makes us space here on the left, because just because we get a critical point, we have to Really show that that value is actually truly emit a right. Let'S do a sign chart and we know we have a critical point at 0 and 0 is really like an end point. You can't get any smaller than 0, but our main critical point is 1.651. So we're going to look at x and see prime sorry, it's not x! Actually w isn't it so let's go and make that a w fix that as a w and then we're gonna. Look at c prime of w. Here we had a little de case, and here we had a 0 okay. So, let's see what happens if we plug in something smaller than 1.65 into our expression, let's plug in say 1. So if we plug in 1 for c prime notice that i will get, if i look at this expression, i will get 40 minus 180 pot. That is definitely negative. Now, let's plug it, something really big well bigger than 1.651 have up to. So if i do plug into that gives me 80 minus 180 over 4 was to a quick calculator just to make sure i do it right. So i get 80 minus 180 divided by 4, and that is a positive, that's still of positive value. Okay, so notice. How, within our possible ranges for w that, since the derivative is going from negative to positive, we definitely have okay excellent. So therefore, but we're not done yet becase, we don't just want the wit. We want the actual minimum cost. So we wanted to take our w value and plug it back in so our c, then at 1.651 is going to equal and let's go ahead and use this 1, because it's right there, okay, so we'll plug in 1.6514, each w so squared plus 180 over 1.651 And what looks like that in our calculator, we get that the cost the cheapest cost for this box is 163.54, so we'll just leave it as dollars and cents all right. So that is the cost of the least expensive box that can be made. According to the recommendation, so that's really great and by the way the w was in meters, so this is actually a pretty big box, so it's really pretty much a huge box anyway. There it is so if we want to help, i do love these application. Problems. Have a wonderful day and see your nme.