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Problem 19 Medium Difficulty

If the farmer in Exercise 18 wants to enclose $ 8000 $ square feet of land, what dimensions will minimize the cost of the fence?

Answer

$x=109.54 \mathrm{ft}$
$y=\frac{8000}{109.54}=73.63 f$

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BC

Blake C.

November 21, 2020

Video Transcript

I what we're told. The farmer and Exercise 18 wants to enclose 8000 square feet of land and were asked to find what dimensions will minimize the cost of defense school. Tomorrow you're calling extra. At 18, we're told that the cost of fencing was $20 per foot. There was no fencing required for the side shirt by the barn little bit and was and the cost of the fencing shared by the neighbor was split in half applause, right? Just left. That's funny. It's just stole. Yeah, that's a good ideas. I mean, I think it was like, opportunity. Who knows? Hilarious. Uh huh. Another podcast. So we'll let the length of the north side of the plot the X. Yeah, and we'll let the length of the west side the plot B Y right? Yeah, she Mm easily. Then we have that to include 8000 square feet of land. X Times y is equal to 8000. Mhm. Yeah, cost of the fence as a function of X and y c f X y. This is 20 times X plus 20 times Why, plus 10 times why for the sides shared with the neighbor, which is 20 x plus 30 y and therefore, as a function of X, we have CFX is 20 x plus 30 times 8000 over X which is 20 X plus 240,000 over X. Now, in order to minimize the cost of the fence, I'm going to need to take the derivative and find critical values. So see, prime of X, this is 20 minus 240,000 over X squared. Find critical values. Want to set this equal to zero? So we have that X squared is equal to 20 over 240,000 Have a I'm sorry. 240,000 over 20 which is yes, 12,000. What's wrong? And therefore X is the square root of 12,000 legislation which is approximately 109 54 Yeah, that would that would be great Trying to figure out exactly. Now we have a second derivative C double prime of X. This is negative. Well, I guess this is positive. 480,000 over X cubed, which we see, of course, is greater than 04 x greater than zero, which is true for our domain. Therefore, it follows that our cost C has an absolute minimum at critical value. X equals 109 54 a belt. Rondo lips poor about the breeders. Mhm, please. So it follows that dimensions that minimize, or we can find why, which is 8000 over approximately 109.54 which is about 73.3 So the dimensions are approximately 109.54 Just what was that? Over. Mm. Okay. And this is indeed. And why is 73.3 also in feet back to