Problem 20 - Precalculus Essentials

Question

The sum of two numbers is \(50 .\) Express the product of the numbers, \(P,\) as a function of one of the numbers, \(x .\)

Step-by-Step Solution

Answer

The product of the numbers, expressed as a function of one of the numbers, is \( P(x) = 50x - x^2 \).

Step 1: Express one number in terms of the other

Let's assume the two numbers are \( x \) and \( y \).Accordingly to the problem, the sum of two numbers is 50. This can be translated into equation as \( x + y = 50 \). Express \( y \) as a function of \( x \) by rearranging the equation, and we get \( y = 50 - x \).

Step 2: Express the product in terms of x

The product of two numbers \( x \) and \( y \) is \( P = x * y \). Substitute \( y = 50 - x \) from step 1 into this product equation. So it will be \( P = x * (50 - x) = 50x - x^2 \). This is the desired function.