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pogonyaev
3 years ago
10

Find two positive numbers satisfying the given requirements. the sum of the first and twice the second is 240 and the product is

a maximum.
Mathematics
1 answer:
MArishka [77]3 years ago
6 0

The two numbers are 120 and 60


Let's call the two numbers x and y. The first sentence translates into


x+2y = 240


from which we can derive


x = 240-2y


So, their product can be written as


xy = (240-2y)y = 240y-2y^2


This expression is a quadratic polynomial with negative leading coefficient, so it represents a parabola concave down. So, the vertex of the parabola is its maximum, which we can find as usual: given the parabola ax^2+bx+c, its extreme point is located at x = \frac{-b}{2a}.


So, in your case, since a = -2 and b = 240, the maximum is located at


y = \frac{-240}{-4} = 60


Now that we know y, we can deduce the value of x:


x = 240-2y \implies x = 240-120 = 120


So, the two numbers are 120 and 60

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