SOLUTION: Find two numbers whose sum is 56 and whose product is as large as possible. [Hint: Let x and 56-x be the two positive numbers. Their product can be described by the function f(x)=x

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: Find two numbers whose sum is 56 and whose product is as large as possible. [Hint: Let x and 56-x be the two positive numbers. Their product can be described by the function f(x)=x      Log On


   

Question 1005533: Find two numbers whose sum is 56 and whose product is as large as possible. [Hint: Let x and 56-x be the two positive numbers. Their product can be described by the function f(x)=x(56-x).]
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Found 2 solutions by stanbon, Alan3354:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
Find two numbers whose sum is 56 and whose product is as large as possible. [Hint: Let x and 56-x be the two positive numbers. Their product can be described by the function f(x)=x(56-x).]
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Equation:
Product = x(56-x)
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P = -x^2+56
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Max occurs when x = -b/(2a) = -56/(2*-1) = 28
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Then 56-x = 56-28 = 28
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Cheers,
Stan H.
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Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Find two numbers whose sum is 56 and whose product is as large as possible. [Hint: Let x and 56-x be the two positive numbers. Their product can be described by the function f(x)=x(56-x).]
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28*28 = 784, the largest possible product.
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x = 28 --> 28*28
Any other combination is (x-a)*(x+a) where a is and integer < 28
(x-a)*(x+a) = x^2 - a^2 which is obviously less than x^2
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The hard way:
f(x) = x*(56-x) = -x^2 + 56x
The max is the vertex at x = -b/2a = -56/-2 = 28
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Using calculus:
f(x) = -x^2 + 56x
f'(x) = -2x + 56 = 0
x = 28 (again)