Find two positive real numbers whose product is a maximum and whose sum of the first number and four times the second is 200 .
A) 160, 10
B) 116, 21
C) 108, 23
D) 100, 25
E) 76, 31

Question

Quizplus · Accepted Answer

To maximize the product of two numbers given a fixed sum, the numbers should be as close to each other as possible. The equation for the sum is $x + 4y = 200$. Rearranging, we get $x = 200 - 4y$. To find the product $P = xy$, we substitute $x$ to get $P = y(200 - 4y)$. Differentiating $P$ with respect to $y$ and setting the derivative equal to zero gives the maximum product. The closest pair to each other from the given options that also satisfies the sum condition is 100 and 25, making option D the correct answer.

Find Two Positive Real Numbers Whose Product Is a Maximum