The sum of two numbers is100. Find the numbers and the maximum product of the square of one by the cube of the other.

Question

Please include your solution on how you arrived with the answer. Thank you!

Answer 1

equation 1:

x+y=100

x=100-y

equation 2:

x²y³=maximize

plug in value for x

(100-y)²y³=(10000-200y²+y³)y³=10000y³-200y⁴+y⁵

0= d/dy(10000y³-200y⁴+y⁵) = 30000y²-800y³+5y4

0=5y²(y²-160y+6000)

y=0 or (y-100)(y-60)=0

or y=60 or y=100

0 and 100 are both invalid answers for y because it minimizes the function rather than maximizing it so the correct answer for y is 60

x=100-60= 40

Therefore, the two numbers are 40 and 60