Find two positive numbers that satisfy the given requirements. The sum of the first number squared and the second number is 54 and the product is a maximum.

36

Step-by-step explanation:

Calculation to Find two positive numbers

First step

Let the first positive number be a

Let the second positive number be b

Second step is to create the Equation

a^2+b= 54 Equation (1)

b = 54-a^2 Equation (2)

Third step is to find the product

Product= a(54-a)^2 = 54a-a^3

Fourth step is to differentiate

dP/dx = 54-3a^2

dP/dx = 0

Fifth step is to solve for 54-3a^2 = 0

Hence,

3a^2=54

a^2 = 18

Last step

a= √(18)

b= 54-(√(18))^2

b=54-18

b = 36

Therefore the two positive numbers are 36

(a) the volume of the cone is 20/3 in3

hence, option a is correct.