00:01
Rancher plans to use 120 feet of fencing and a side of his barn to form of rectangular boundaries for cattle.
00:08
So what dimensions of the rectangle would give the maximum area and what's the area? so remember that only three sides require fencing because one side is of his barn.
00:18
So there is no fencing required for that side.
00:20
Let's say over here we have barn and these sides are x and this is y.
00:27
So total fencing available will be same as the total.
00:30
Length of these sites because this is same as the fencing which is required so x plus x plus y will be 1200 feet which means that 2x plus y is 1200 feet and from here we can get the value of y as i'm sorry i was saying 200 that's 1200 1200 minus 2x so that's call it equation 1 now we'll talk about the area which we want to maximize so the area of a rectangle is definitely x times y so x times y is the area and we already have and since we have to maximize the area so maximizing a thing would mean that in a function we have to differentiate equate to zero find critical points and then find the maximum so in order to differentiate this it's always a good idea to have the function only in terms of x and we do not mean y over here and so we can eliminate y from equation one because we have y in terms of x so replacing y as 1200 minus 2x this will become a as 1200 x minus 2x square so a prime is going to be 1200 and the differentiation of 2 comes out and the differentiation of x square is 2x over here so this will become a prime will be 1200 minus 4x this means that a prime should be 0 which means that 1200 minus 4x should be 0 which means that 1200 should be equal to 4x and dividing both sides by 4 we have 300 is equal to x but the question is whether it has a maxima or minima so that's where a double prime comes in so if you find a double prime from here we will be left with minus 4 and since a double prime is negative this would mean that the critical point has a maxima that is the that is the check which we do so x equal to 300 will have a maximum corresponding y will be 1200 minus 2x so y is 1200 minus 2 times x x x is 300 so that's 1200 minus 600 which is 600 that is the value of y.
02:45
We got x we got y and we need to find the area as well.
02:49
So area is just x times y which will be the maximum area.
02:53
So the area maximum is just going to be x times y.
03:00
So that's 300 times 600, which is 18 followed by four zeros over here.
03:08
Only thing we need to take care is about the area...