Hello, Luhan,
Let's set W for the width and L for the length. The perimeter will be 2 times the width pus only 1 times the length, since one length edge is bounded by the creek. (The assumption here must be that no one dares getting their feet wet). We'll say the total perimeter length is P.
We can state:
P = 2W + 1L
We also know that W*L is the area and must be equal to 60m2
W*L = 60m2
We can use the second equation in the first, by rearranging and substituting:
L = 60/W
P = 2W + 60/W
I'll rewrite this with x and y, to make it more familiar:
y = 2x + 60x-1
This is an odd equation to graph, but it will show a curve with a sharp decline as it moves to higher values of x from the origin. The origin, x=0, is undefined, since the 60/x term is divided by zero. By once we get above around x=1, the graph suddenly slows it's descent and eventually makes it's way to a minimum, before slowly rising once again. Y is the perimeter. x is the length. Everywhere on the line represents an enclosure of 60m2. So any point on the line will satisfy the pen's area requirement. But we want the shortest perimeter (y), which seems to be around 5, or so.
An easier, and more accurate, method to find the minimum is to note that the slope of the line is 0 at the minimu, so if we take the first derivative of the equation and set it to zero, we can calculate the minimum for the perimeter.
y = 2x + 60x-1
y' = 2 + (-60x-2)
0 = 2 + (-60x-2)
0 = 2x2 - 60
x2 = 30
x = 5.477 meters
We now have W= 5.477 meters
That means L, the length, is 60/W or 60/5.477
L = 10.95 meters
W*L is supposed to be 60m2
(10.95m)*(5.477m) = 60m2 It checks
The total perimeter needed is 2W+L=21.91 meters
Interesting problem. I hope I did it correctly;
Bob
