If 550 m of fencing is available to build five adjacent pens, as shown in the diagram on the left, express the total area A of the pens as a function of x. What value of x will minimize the total area? What is the maximum area?
describe the diagram.
Whatever the layout, I'm sure the maximum area will be when the fence is divided equally among lengths and widths.
Here is the layout,
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Height is x and width is y
To express the total area A of the pens as a function of x, we need to determine the dimensions of each pen in terms of x.
Looking at the diagram, we can see that each pen consists of a rectangle and a semicircle attached on the ends. The width of each rectangle is x, and since there are five pens, the total width would be 5x.
The length of each rectangle, on the other hand, is derived from the remaining length of the fence after forming the semicircles. If we subtract the length of each semicircle from the total length (550m), we can divide the remaining length by 5 to get the length of each rectangle.
The length of each semicircle is determined by half the circumference, which is πr, where r is the radius. In this case, the radius would be x/2 since it connects to a side of the rectangle. Therefore, the length of each semicircle is π(x/2).
Subtracting the total length of the semicircles from the total available length gives us:
550m - 5π(x/2).
Dividing this remaining length by 5 to get the length of each rectangle, we have:
(550m - 5π(x/2)) / 5.
Now, the area of each rectangle is given by the product of the width and length, which is x * [(550m - 5π(x/2)) / 5].
Since there are five identical pens, the total area A would be the area of one pen multiplied by five:
A = 5 * x * [(550m - 5π(x/2)) / 5].
Simplifying this expression, we get:
A = x * (550m - 5π(x/2)).
To find the value of x that minimizes the total area, we can differentiate the function A with respect to x and set the derivative equal to zero. Then, solve the resulting equation for x.
To find the maximum area, you can substitute the value of x that minimizes the area into the function A, and that will give you the maximum area.