Max is building a rectangular dog pen. One side of the pen will just be the side of the house. If he has 45 feet of fencing to use for the other three sides, what is the largest area he can create? Round to the nearest tenth.

Question

Max is building a rectangular dog pen.  One side of the pen will just be the side of the house.  If he has 45 feet of fencing to use for the other three sides, what is the largest area he can create?  Round to the nearest tenth.

anonymous · Answer

@imqwerty

biohazard9064 · Answer

well what do you think you start with?

anonymous · Answer

The largest rectangular area is a square. Since the sides of a square are equal, just divide 45 by 3 to find the length of one side. Area of a square is side times side.  Right ??

biohazard9064 · Answer

yep

anonymous · Answer

But because it says round to the nearest 10th i thought it was wrong because it doesnt make sense .

anonymous · Answer

So its like 15x15 ??

biohazard9064 · Answer

i believe so

biohazard9064 · Answer

hold on

anonymous · Answer

Okay

biohazard9064 · Answer

Yes I believe its 15 x 15 is it an answer choice or fill the blank?

anonymous · Answer

fill in the blank .

biohazard9064 · Answer

So i would assume its 15 x 15 it makes sense but im gonna have @imqwerty double check

anonymous · Answer

Thank you

mathmale · Answer

Is this problem from algebra or from calculus?

Your best bet would be to write the equation for the area of the rectangle.  Because one side of the rectangle does not actually require fencing, it's not appropriate to assume that a square field would have the max area possible.

Draw the field.  Draw the 45 degree angle.  Come up with a formula for the area of this field as a function of x.  Graph this equation.  It will likely rise, then fall, as x increases.  Estimate the x value at which the area is at its max for this situation.

anonymous · Answer

Algebra

mathmale · Answer

Then drawing a graph of the area function would be your best bet.

imqwerty · Answer

have been told this in class-> The largest rectangular area is a square. 
?
or you just took it yourself?

anonymous · Answer

I googled it

imqwerty · Answer

okay we do it step by step :)
you answer the question i ask

anonymous · Answer

Okay Cool

imqwerty · Answer

okay we take it to be a rectangle of side lengths $x$ and $y$

now tell what will be the perimeter of this rectangle

anonymous · Answer

But i dont knoow the lengths ??

imqwerty · Answer

yes we don't know but we assumed the side length to be $x$ and $y$
just write the perimeter in terms of $x$ and $y$

anonymous · Answer

x+x +y+y ???

imqwerty · Answer

yes correct :)
so we can write it like this->
$P=2x+2y$ 
okay?

anonymous · Answer

Okaay

imqwerty · Answer

now try to isolate y :)

anonymous · Answer

do i just divide by 2 ?? or do i have to like move it to the other side first

anonymous · Answer

sorry im kinda stupid

anonymous · Answer

It's fifteen.

imqwerty · Answer

no its not
please refrain from giving direct answers :)

imqwerty · Answer

okay to isolate y just do this->
$2x+2y=P$
$2x+2y-2x=P-2x$
$2y=P-2x$
$y=\large{\frac{P}{2}} -x$

imqwerty · Answer

okay?
here we are trying to prove that rectangle of greatest area is a square with the help of an example..

anonymous · Answer

Yeah

imqwerty · Answer

okay now tell what will be the Area of this rectangle

mathmale · Answer

The greatest possible area of a rectangle with four sides made up by fencing is A=x^2, where x is the length of one side.  Yes.

But if you need to form an area when one of the sides does not require fencing, the greatest possible enclosed area is NOT A=x^2.

mathmale · Answer

Again, I suggest you draw a picture and come up with a formula for the enclosed area.  Then, graph this formula.  At which x value do you obtain max area?

imqwerty · Answer

yea drawing the graph is more easier approach :)

mathmale · Answer

Yes, AFTER you have developed a formula for the area in terms of one variable, x.

anonymous · Answer

Ill do that thanks guys .