Answer to Question #206746 in Calculus for Cess

Question

The product of two numbers is 4 square root of 3. Find the numbers so that the

sum 𝑆 of the square of one and the cube of the other is as small as

possible

Answer 1

let x and y be two numbers.

The product of two numbers,

$xy=4\sqrt3\\ x=\frac{4\sqrt3}{y}$-----------(1)

sum 𝑆 of the square of one and the cube of the other is as small as

possible

$S=x^2+y^3\\ =(\frac{4\sqrt3}{y})^2+y^3\\ =\frac{48}{y^2}+y^3$

Here, S is the objective function.

$S'=\frac{-48}{y}+3y^2\\ Now, S'=0\\ \frac{-48}{y}+3y^2=0\\ y=2.5 \implies x=\frac{4\sqrt3}{2.5}=2.7$

thus, x=2.7 and y=2.5