Answer: the numbers are 5,111.3 and 35.74
Step-by-step explanation:
I will answer this in English.
We have the numbers X and Y, ad we have that:
X + 2*Y^2 = 7776
X = 7776 - 2*Y^2
Now we also want that the product of X and Y is a maximum.
The equation of maximum product is
P(Y) = X*Y = (7776 - 2*Y^2)*Y
= 7776*Y - 2*Y^3
Now, to find the maximum we must derivate and find the value where P'(Y) is equal to 0
P'(Y) = 7776 - 6*Y^2 = 0
7776 = 6*Y^2
Y = √(7776/6) = 35.74
Now, we replace this value in the equation for X and we can find X:
X = 7776 - 2*Y^2 = X = 7776 - 2*(35.74)^2 = 5,111.3
