What are two positive real numbers whose product is a maximum and whose sum of the first number and four times the second is 160?

Let the two positive real numbers be ( x ) and ( y ). The constraint given is ( x + 4y = 160 ). To maximize the product ( P = xy ), we can express ( y ) in terms of ( x ): ( y = \frac{160 - x}{4} ). Substituting this into the product gives ( P = x \left(\frac{160 - x}{4}\right) = \frac{160x - x^2}{4} ), which is a quadratic function in ( x ) that opens downwards. Maximizing ( P ) occurs at the vertex, ( x = 80 ), leading to ( y = 20 ), thus the numbers are ( 80 ) and ( 20 ).