Arnold has 24 m of fencing to surround a garden, bounded on one side by the wall of his house. What are the dimensions of the largest rectangular garden that he can enclose? I am having problems solving this. I heard the starting step is y=x(24-2x) but i am very confused. Please can someone explain step by step.

Question

anonymous · Answer

that's how you're suppose to do it, but here's a hint. With problems like these where you have a fixed perimeter, you will always get the largest area when your fence is a square ;) 

that should make it easy for you :P

anonymous · Answer

Okay im following with you. Where did you get 24 = 2y + 2 at the beginning? where'd the x value go?

anonymous · Answer

sorry, it was a typo. there should be an x there, it comes back after the  arrow ----> 

i still worked it out correctly, just pretend it's there

anonymous · Answer

Ah sorry. I think you made a mistake. The question says the fence is bounded by one side by a wall of the house. So there's only one width or length value. So it'd be 2y + x = 24. This is where I get stuck cause I know there's a quadratic equation that's supposed to form but I'm shgsjhgjsh trying to get it

anonymous · Answer

ah snap, i didnt even see that part :P

anonymous · Answer

still, a square would give you the biggest area!

anonymous · Answer

Yep I'm trying to work it out right now following your steps but with the fix. I'll post back in a min and see if I got the proper answer

anonymous · Answer

okay let me start over haha

anonymous · Answer

24 = y+2x  (comes from perimeter)

y = 24 - 2x

A = x*y,   and y = 24 - 2x so we get ---> A = x(24-2x) ---> A = -2x^2 + 24x

now we can see that the Area equation is a downward parabola right?





so now we take the derivative of A and set it equal to 0, so we can solve for what value of x is at the maximum point of the parabola



then just like before, we take that value for x and plug it back into the equation
y = 24 - 2x to solve for what y is

anonymous · Answer

A plot and solution using Mathematica 8 Home Edition is attached.
Notice that the enclosed garden is a rectangle, not a square.

anonymous · Answer

A square garden of 8 meters on a side is 64 square meters.  Mathematica's rectangular garden of 6 by 6 by 12 meters encloses 72 square meters of garden.