Problem 62 - Algebra and Trigonometry

Question

Among all pairs of numbers whose sum is \(20,\) find a pair whose product is as large as possible. What is the maximum product?

Step-by-Step Solution

Answer

The pair of numbers whose sum is 20 and whose product is as large as possible is 10 and 10. The maximum product is 100.

Step 1: Write the product in terms of one variable

Given the constraint that x + y = 20, you can express y in terms of x, that is y = 20 - x. Then, the product p(x) becomes p(x) = x * (20 - x).

Step 2: Find the vertex of the parabola

The function p(x) = x * (20 - x) is a parabola and its maximum occurs at the vertex. The x-coordinate of the vertex of a parabola given as y = ax^2 + bx + c is -b/2a. In this case, rewriting p(x) in the standard quadratic form gives p(x) = -x^2 + 20x. Comparing this with the standard form, we have a = -1 and b = 20. Thus, the x-coordinate of the vertex is -b/2a = -20/2*(-1) = 10.

Step 3: Compute the maximum product

Substitute x = 10 into the equation for p(x) to get the maximum product. This is p(10) = 10 * (20 - 10) = 10 * 10 = 100.