We're stuffing a crash course of calculus into the last few weeks of my math class and I'm terrible at this new stuff, and we have a quiz on it tomorrow. The teacher's tried explaining it to me, but it doesn't help. This one problem makes no sense. It's optimization.
A farmer plans to fence a rectangular pasture adjacent to a river. The pasture must contain 180,000 square meters in order to provide enough grass for the herd. What dimensions would require the least amount of fencing if no fencing is required along the river?
I know the answer, but I have absolutely NO idea how it got there. If anyone was actually good at this sort of thing, I'd love you forever. The answer's 600x300 meters, but I can't get anything from what I try.
I'm outside your picture frame, and the glass is breaking now...
well i don't know if this will help but you could use your tables so find the factors of 180 000 or 18 cause it's easier so 1 x 18, 2 x 9 or 3 x 6. then add each of them together so 19, 11 and 9. then as 9 it the smallest it uses less fencing. and you get the 9 of fencing from 3 x 6 so putting the 0's back in 300 x 600.
hope it helps a bit i'm not great at explaning things
It could work, but tomorrow I'm supposed to find it using the derivative, and this entire thing is stupid. I've never been good at optimization, and the calculus just makes it more confusing. Thanks though!
I'm outside your picture frame, and the glass is breaking now...
oooo I remember doing an example so similar to this last year...why do they always use farmers and fenced...? Hmm...
Anyway...it's basically a simple differentiation question that they put into silly words and make it look confusing...
It helps to start out with a diagram so you know what you're working with
Do you have any more details about the question? A diagram? Or something we can put into an equation?
The only time you will find real light is when you're searching in the dark..
that's all the problem gave me. So far I know the fence must equal 2x+y, because theres two xes and only one y, since we have the river. So 180,000=xy, so y=180,000/x. So then F= 180,000/x +2x. The derivative is f'=2-180,00/x(squared). So the critical point would be x=0, but that doesn't make any sense in the problem and it doesn't give the answers in the back of the book.
I'm outside your picture frame, and the glass is breaking now...
Eeep...ok I've been sat here trying to work it out but my brain's just completely shutting down...I'm so sorry...I'll keeep trying though and will let you know if I get there
The only time you will find real light is when you're searching in the dark..
It's ok don't worry about it maths used to be my whole life at one point...[sad, I know]. It was always optimization that we all struggled with though...
The only time you will find real light is when you're searching in the dark..
Yeah, I was good with limits. I thought calculus was easy, then we got into this stuff and I realized she'd just been waiting to dump this on. Even worse, we've got related rates next week. I can't wait, there's only next week and then the final.
I'm outside your picture frame, and the glass is breaking now...
Let x be the length of the sides perpendicular to the river, and let y be the length of the side across from the river.
Start with x*y = 180,000
Then, y = 180,000/x
Now let P be the perimeter of the rectangle (i.e. the total length of the fence). That's the thing you're trying to minimize.
P = 2x + y
P = 2x + 180,000/x
Now you just need to take the derivative of P with respect to x, set that equal to 0, and solve for x.
dP/dx = 2 - 180,000/(x^2) = 0
Solve for x. You should get x = 300.
Now go back to the original equation, x*y = 180,000, and solve for y. You should get y = 600.
So the dimensions are 300m x 600m, and the longer side is parallel to the river.
Emily
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...I think I just kept forgetting the two. Wow, I feel stupid. Thanks!
I still dont know how I'll do on the quiz because my teacher started every problem, but hey, at least I'll know how this one's done. Maybe she'll pick easy problems.
I'm outside your picture frame, and the glass is breaking now...
...I think I just kept forgetting the two. Wow, I feel stupid. Thanks!
Don't feel stupid - I still do things like that all the time, and I'm in college and just finished multivariate calculus.
But when you were coming up with x = 0 earlier, it's also important to remember that you'd have had to divide by 0 to get that. If 180,000/(x^2)=0, then there are no solutions because you'd have to multiply both sides by x^2, and you'd end up with 180,000 = 0.
Emily
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