Find two positive real numbers whose product is a maximum.
The sum of the first and three times the second is 48.

A)24,8
B)24,16
C)16,4
D)24,6
E)42,6

Question

Find two positive real numbers whose product is a maximum. ​
The sum of the first and three times the second is 48.
​
A)24,8
B)24,16
C)16,4
D)24,6
E)42,6

Quizplus · Accepted Answer

Given the constraint that the sum of the first number and three times the second number is 48, we can express this as x + 3y = 48. To maximize the product xy, we can use the method of Lagrange multipliers or simply express one variable in terms of the other and substitute into the product expression. Solving x + 3y = 48 for x gives x = 48 - 3y. Substituting into xy gives the product as a function of y: (48 - 3y)y. Differentiating this product with respect to y and setting the derivative equal to zero can help find the value of y that maximizes the product. However, without calculus, we can reason that since the sum of x and 3y is constant, the product xy is maximized when x and y are as close to each other as possible due to the arithmetic mean-geometric mean inequality. The closest x and y can be under the given constraint is when x = 24 and y = 8, which satisfies x + 3y = 48 and maximizes the product xy.

Find Two Positive Real Numbers Whose Product Is a Maximum