Problem 62 - College Algebra

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 maximum is (10,10), and the maximum product is 100.

Step 1: Set up the formula for the sum of the two numbers

We can represent the two unknown numbers as \(x\) and \(y\). Since we know that they sum up to 20, we write the formula \(x + y = 20\).

Step 2: Solve above equation for one variable

Let's solve for \(y\) in terms of \(x\). This means we have \(y = 20 - x\).

Step 3: Express the product formula and substitute

The product of the two numbers is given by \(xy\). So, substituting \(y\) in terms of \(x\), we have \(x * (20 - x) = 20x - x^2\). We find that this formula represents a downward-facing parabola. The maximum value of this product will occur at the vertex of the parabola.

Step 4: Find the maximum of the product function

The maximum value for a downward-facing parabola \(ax^2 + bx + c\) occurs at \(x = -\frac{b}{2a}\). In our case, \(a = -1\) and \(b=20\). Therefore, the maximum occurs at \(x = -\frac{20}{2*-1} = 10\).

Step 5: Find maximum product

Substitute \(x = 10\) in the product \(20x - x^2\) to find \(20*10 - 10^2 = 200 - 100 = 100\) as the maximum product.