Answer to Question #249611 in Calculus for katlyn h. degamo

Question #249611

The product of two positive numbers is 4 √ 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. 


1
Expert's answer
2021-10-12T14:29:54-0400

First, we have the following relation that will help us to solve the equation that has to be minimized:


"xy=4\\sqrt{3} \\implies x=\\cfrac{4\\sqrt{3}}{y}; x>0, y>0"


Then, we use the formula for the sum and we substitute x to find an equation in terms of the other variable (y)

"S=x^2+y^3= \\Bigg( \\cfrac{4\\sqrt{3}}{y} \\Bigg)^2+y^3= \\cfrac{48}{y^2}+y^3\n\\\\ \\cfrac{dS}{dy} = -\\cfrac{96}{y^3} +3y^2"


To find the minimum we have to make "\\cfrac{dS}{dy} =0" and then solve for y:

"-\\cfrac{96}{y^3} +3y^2=0\n\\\\ \\implies 3y^2=\\cfrac{96}{y^3} \\, \\therefore y^5=\\cfrac{96}{3}=32=2^5\n\\\\ \\implies y=2\n\\\\ \\therefore x= \\cfrac{4\\sqrt{3}}{2}=2\\sqrt{3}"


In conclusion, the numbers "x=2\\sqrt{3}" and "y=2" satisfy the relation given ("xy=4\\sqrt{3}" ) and the value for S is minimum.

Reference:

  • Thomas, G. B., & Finney, R. L. (1961). Calculus. Addison-Wesley Publishing Company.

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