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Chapter 3: Problem 62

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

Short Answer

Expert verified
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 by step solution

01

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).
02

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.
03

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.

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