The function f(x) = –(x – 3)2 + 9 can be used to represent the area of a rectangle with a perimeter of 12 units, as a function of the length of the rectangle, x. what is the maximum area of the rectangle?

Step-by-step explanation:

f(x) = –(x – 3)2 + 9

      = 3² - (x – 3)2  identity a²-b² = (a+b)(a - b)  where : a =3 and b = x-3

     = ( 3+x-3) (3-x+3)

    f(x) = x( 6-x)

the perimeter is ; 2(x +6-x) =12

the maximum area of the rectangle is : f(3) = 9 m²

The maximum area of rectangle is:

                   9 square units

The function f(x) is given by:

where f(x) represent the area of rectangle.

Also, the area is maximized at the value of x where the derivative is equal to zero.

Hence,

Hence, when x=3 the area of the rectangle is maximum.

Also, the maximum area of rectangle is:

step-by-step explanation:

12: 00 p.m. to 7: 00 p.m. is a total of 7 hours.

4: 00 p.m. to 7: 00 p.m. is a total of 3 hours.

this means that from 4: 00 p.m to 7: 00 p.m. this represents 3/7 of what was lost from 12: 00 p.m. to 7: 00 p.m.

after knowing this, after doing 11,600 - 6,800 = 4,800

4,800 represents 3/7 of what was lost for 7 hours.

we then divide 4,800 by 3 which gets 1,600.

because we know that 1,600 represents 1/7 of how much water was lost, we just have to multiply by 7, which gets 11,200.   after adding 11,200 to how much water was left at 7: 00 p.m. which was 6,800. 11,200 + 6,800 = 18,000

this means that the pool originally had 18,000 gallons in the pool.

you can also check your work by doing 18,000 - 4(1600), which becomes           1800 - 6400 which gets you 11,600.