the sum of two positive numbers is 800, what should the number be if their product is as large as possible? first number: second number:

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violy | High School Teacher | (Level 1) Associate Educator

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Let set x and y be the numbers. 

Taking note that the sum of the numbers is 800. 

Our first equation will be x + y = 800. 

Solve for x in terms of y, subtract both sides by y. 

x = 800 - y.

Let set P = the product of the two numbers. 

So, we will have: P = xy. 

Plug-in x = 800 - y. 

P = (800 - y)(y) 

Use Distributive property. 

P = 800y - y^2 

Take the derivative. 

P' = 800 - 2y. 

Equate the zero for the critical number. 

800 - 2y = 0 

Add 2y on both sides. 

800 = 2y 

Divide both sides by 2. 

y = 400 

Take the second derivative P'' = -2. hence, we have a mximum at y = 400. 

Solve for x = 800 - 400 = 400. 

Therefore the largest possible number that will create the largest product are 400 and 400

 

 

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