Find two positive numbers such that the sum of the first and twice the second is 96 and whose product is a maximum.
A)48 and 24
B)44 and 26
C)42 and 27
D)48 and 48
E)54 and 21

Question

Quizplus · Accepted Answer

The problem can be solved using the method of Lagrange multipliers or by setting up an equation based on the given conditions and then differentiating to find the maximum product. Let the two numbers be x and y. According to the problem, x + 2y = 96, and we want to maximize the product P = xy. Solving the first equation for x gives x = 96 - 2y. Substituting this into the product equation gives P = (96 - 2y)y. Differentiating P with respect to y and setting the derivative equal to zero gives the values of y that maximize or minimize P. Solving this will lead to finding that the product is maximized when x = 48 and y = 24.

Find Two Positive Numbers Such That the Sum of the First