what are the maximum dimension how do i find them

Question

i know the area is 40 but how do i find the maximum dimensions?

 

 

 

A homeowner is building a pen for his dog and wants to maximize the area. He purchases 40 feet of fencing. If he builds a rectangular pen, what dimensions would maximize the area? What would the area be?

Answer 1

The area is NOT 40 feet. Since it is a rectangle with a perimeter of 40 feet, the length of one width and one length is 20 feet. Let x be the width. Then 20-x will be the length. The area is now x (20-x) or 20x - x^2.

 

Now the area is found by finding the first derivative of the function and setting it equal to zero.

 

The first derivative of 20x - x^2 is 20 - 2x.  If 20 - 2x = 0 then x =10.

 

Then the width is 10 and the length is 20 - 10 = 10 also.

 

You can maximize the area by creating a square with each side 10 feet. This will make the area 100 square feet.

Answer 2

40 feet of fencing means perimeter is 40

2(l+b) = 40

l+b = 20

 

make a table

l= 10 b= 10 area = 100

l=11 b = 9 area = 99

l=12 b=8 area = 96

l=13 b = 7 area = 91

l= 14 b =6 area = 84

l=15 b = 5 area = 75

l=16 b=4 area = 64

l=17 b = 3 area = 51

l=18 b=2 area = 36

l= 19 b=1 area = 19

 

then if l=10, 1 2 3 ,4.... and b=10, 19, 18, 17 ... it'll be the same calculations as in the table

As you can see for l+b = 20, when l=10 and b = 10 yields the maximum area of 100

 

Hope this helps