a rectangular pen is divided into 6 rectangular strips by fences running parallel to the width of the pen. if 420m of fencing are available, determine the dimensions that give the maximum area possible.
To determine the dimensions that give the maximum area possible for the rectangular pen, we need to set up an equation and differentiate it to find the critical points.
Let's assume the width of the rectangular pen is x. As mentioned in the problem, the pen is divided into 6 rectangular strips by fences running parallel to the width. This means that there are 7 fences in total. Since there are 7 fences, the length of each rectangular strip will be 420m/7 = 60m.
Now, we can calculate the dimensions of the pen. The width of the pen is x, and the length of each rectangular strip is 60m, so the length of the pen will be 60m * 6 = 360m.
The area of the rectangular pen can be calculated by multiplying the width (x) with the length (360m):
Area = x * 360m
To find the value of x that gives the maximum area, we need to differentiate the equation and set it equal to zero:
d(Area)/dx = d(x * 360m)/dx = 360m
Setting this derivative equal to zero, we get:
360m = 0
This equation has no solution because 360m is a constant.
Since there is no critical point, the area will either be increasing or decreasing indefinitely, which means there is no maximum area for this rectangular pen with the given constraints.
Therefore, there is no specific set of dimensions that give the maximum area possible.