There are two positive numbers The square of the first numbe

There are two positive numbers. The square of the first number, plus five times the second number, is 900. Their product is a maximum. Find the two numbers.

Solution

there are x and y numbers,

so,

x^2 + 5y = 900

so ,

y = (900-x^2)/5..................(1)

P = xy

P = x * (900 - x^2 ) / 5

P = (180 x - x^3/5)

product is maximum, so P\' = 0

P \' = 0 = (180-3x^2/5)

180 = 3x^2 / 5

x^2 = 300

x = 10 sqrt(3) or -10sqrt(3)

taking in equation (1)

y in both case taking x^2 = 300

y = (900-300)/5

y = 120