Question

Please help: A rectangular pen is to be built with a 1200 m of fencing. the pen is to be divided into three parts using two parallel partitions.
find the max. possible area of the pen.

Answer 1

Note here that the amount of fencing must be 1200m and that we are trying to maximise the area by keeping the same perimeter of the pen but changing the dimensions. So if we have a pen like this:The perimeter of this rectangle with sides x and y will be:\[\bf Perimeter = 2x+2y=2(x+y)=1200\]And since we are already aware of the perimeter, we can right away say that it's 1200.

Now the area will be given by:\[\bf Area=x \times y\]Notice how we have 2 variables here. It would be much easier to maximise the area if it was given in terms of a single variable. So may be can substitute one variable from given Perimeter equation?\[\bf 2x+2y=1200 \implies y = \frac{1200-2x}{2}=600-x\]We can substitute this value of y in to our Area equation:\[\bf Area=x \times y = x(600-x)=-x^2+600x \]Now look at that, we have the Area as a quadratic! We also know that this parabola opens downwards since it has a "- sign" in front of "x^2" so it must have 1 single maximum value which occurs at the vertex, and this must be the largest value the area can achieve such that the perimeter is 1200 m. We know that the vertex always occurs exactly in between the zeroes/x-intercepts of the parabola which is the same as saying that the x-value of the vertex is -b/2a, where "b" is the coefficent of "x" and "a" is the coefficient of "x^2". Since this parabola has x-intercepts at x = 0 and x = 600, the vertex will occur right in between these two points so it will be at:\[\bf \frac{ -b }{ 2a }=\frac{ -600 }{ -2 }=\frac{ 600-0 }{ 2 }=300\]So the maximum of this area function occurs at x = 300, which is the x-value of the vertex of the parabola. To find the value of the area at this x-value, we compute the value of the function at 300:\[\bf f(300)=-(300)^2+600(300)=-90000+180000=90000\]Therefore, the maximum area achievable with a perimeter of 1200m is 90000 m.

@0202 (Note you could also use derivatives to find the maximum of the quadratic but a non-calculus approach is always nice [= )

Answer 2

the answer in the textbook is actually 45,000 m^2., and where did u get the -b/a equation from? @genius12

Answer 3

oh wait i think i see the problem...we didn't take into account that its divided into 3 parts so the first equation should be 2x+2w=1200

Answer 4

45,000 is incorrect. It has to be 90,000 m^2. Your textbook probably made a mistake.

Answer 5

the question, the way you wrote it, asks for the area of the pen as a whole, not of the partitions.

Answer 6

\[1200= 2l +4w \] you take that equation and solve for w which will give you \[300-\frac{ 1 }{ 2 }l = w \] then you plug it into the equation A=lw and you'll get \[A = 300l-\frac{ 1 }{ 2 }l ^{2}\] and you take the derivative of that and get l=300. take l=300 and plug it back into the perimeter equation which will give 150=w. so to get the max area u just A=300x150 = 45 000