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Problem 17 Medium Difficulty

Do Exercise 16 assuming the container has a lid that is made from the same material as the sides.

Answer

$\$ 191.28$

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Video Transcript

you don't like were asked to do exercise 16, assuming the container has a lid that is made from the same material as the sides. So recall that an exercise 16 of this section. Yes, I like I like something the world is about to hit. Reset. Mm. Civilian were given open box container with a volume of 10 the length that's twice its width. And the cost is 10 per square meter of the basin, six per square meter of the sides. And we were asked to find the minimum cost of the box. Yeah, yeah. Mhm. Yeah. So once again, we'll let w b the width forget of the base Adams. And to w is the length of the base. I mean, I don't even know how to qualify. Yes. Oh, h is the height of the box. How and the volume, which we're told is 10. This is also equal to base with times base length. So w times to w times the height H which is two w squared h. This is in cubic meters. What was now? I want to solve the volume for H. So we have a choice. Is equal to 10 over to W squared, which, of course, is five over w squared the cost of the base. Well, this is going to be There's this girl named Oh, I know. And she had me 10. It's $10 per square foot times the area of base, which is to W times W, which is 20 w squared. Yeah, and the cost of the size of the lid Look. And she was standing, Yes. Yeah, and I was loved out of being just he did is the cost per square meter, which is six mm times the area of two sides with base w So this is two times w times h plus mm area of two sides of days to W. This is plus two times two w times h plus the area of the lid, which is the same as the area of the base to W times W. And it's like this this son, and this simplifies to wasn't right. So we have to w h plus four wh plus two w squared three of six times six is 36 w h plus 12 w squared, Mr Right, and therefore the total cost c. This is the cost of the base plus the cost of sides and lid, which is 36 w h plus 32 w squared. Now substitute eight from H from our volume equation. So this is actually so 36 w don't seem like a gay bridge. Times five over w squared You were almost like missed that plus 32 w squared. And so, as a function of W C is equal to 36 times five, you know what which is 180 over W plus 32 w squared. I hate I like them. Therefore, the derivative of C with respect to W, this is negative. 180 over W squared plus 64 w To find the critical values we set this equal to zero not illegal we're discussing. So we have 64 w equals 1 80 over w squared solving for W. We have w cubed equals 45/16 and therefore W is equal to the cube root of 45/16, which is approximately 1.41 one 1.412 I guess. Yeah. Rick Astley British. Now we can find the second derivative Cedo Prime of W. This is, let's see positive. 360 over W Cubed plus 64. And it follows that C double prime of 45/16 to the one third is automatically positive. So by the second derivative test, there was a weird like research when I started. It follows that we have an absolute Percy has an absolute minimum at W equals 45 16th to the one third. Yeah, lot of guys basically, and the minimum cost is C of 45 16th to one third. Plug this in. This is approximately 191.28 here, and this is in dollars said Not like that.