Question 1001517
your steps are:


I think the answer goes:
A = 2x*y
A = 2x*(-x^2+9)
A = -2x^2 + 18x ****************************************


A = -2x^2 + 18x is where you went wrong.
2x * (-x^2 + 9) is equal to -2x^3 + 18x.
find the derivative of that and you get -6x^2 + 18.
set that equal to 0 and you get -6x^2 + 18 = 0
add 6x^2 to both sides of that and you get 6x^2 = 18
divide both sides of that by 6 and you get x^2 = 3
take the square root of that and you get x = plus or minus sqrt(3).


the value of x is plus or minus sqrt(3).


the height of your rectangle is y = 9-x^2
the width of your rectangle is 2x.
the maximum area is when x = plus or minus sqrt(3).
when x = sqrt(3), your area is (9 - sqrt(3)^2) * 2 * sqrt(3)) which is equal to 20.78460969.


you can also find this graphically by graphing y = (9 - x^2) * 2 * x.


the graph of that equation is shown below:


that's the graph of the area of your rectangle and it shows that the maximum area is when x = 1.732 which is the rounded version of sqrt(3).





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