# How do you find two positive numbers whose product is 750 and for which the sum of one and 10 times the other is a minimum?

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Let ( x ) and ( y ) be the two positive numbers. We are given that ( xy = 750 ). We want to minimize the function ( f(x) = x + 10y ). To find the minimum, we can use the method of substitution. Solving for ( y ) in terms of ( x ) from the equation ( xy = 750 ), we get ( y = \frac{750}{x} ). Substituting this expression for ( y ) into the function ( f(x) ), we get ( f(x) = x + 10\left(\frac{750}{x}\right) ). To minimize ( f(x) ), we differentiate with respect to ( x ), set the derivative equal to zero, and solve for ( x ). Differentiating ( f(x) ) with respect to ( x ), we get ( f'(x) = 1 - \frac{7500}{x^2} ). Setting ( f'(x) = 0 ) and solving for ( x ), we find ( x = \sqrt{7500} ) or ( x = -\sqrt{7500} ). Since ( x ) must be positive, we take ( x = \sqrt{7500} ). Substituting this value of ( x ) back into the expression for ( y ), we find ( y = \frac{750}{\sqrt{7500}} ). Therefore, the two positive numbers are approximately ( x \approx 27.39 ) and ( y \approx 27.39 ).

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