\"A length of wire fencing is 20 meters long. You and a friend both have dogs, a

Question

"A length of wire fencing is 20 meters long. You and a friend both have dogs, and decide to use this wire fencing to make both a circular enclosure and a square enclosure for your two dogs. Of course, you could make just one enclosure(either circular or square) and let the dogs play together, but your friend argues that if you make two enclosures, you'll get a larger total area for your pets. So, you decide to use x meters of fence for the circle, and the rest for the square." So this was the problem.. The questions are: A) Find the total area of both enclosures if x=8 B) Find the value of x that gives you the largest total area. C) Find the value of x that gives you the smallest total area. Pleaseeeee! I really need help on this.

Explanation / Answer

a) wire used =2*pi*r = 8 => r =1.27

area = 5.09 m2

area of square = 9m2 so total area = 14.09 m2

b) radius of circle = x/(2*pi )

area of circle =x^2/(4 )

area of square = ((20-x)/4)^2

total area = ((20-x)/4)^2 + x^2/(4 )

for maximization da/dx =0

x/2 = 1/8 *(20-x)

so x= 4

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