Problem 61 - Algebra and Trigonometry

Question

Among all pairs of numbers whose sum is \(16,\) find a pair whose product is as large as possible. What is the maximum product?

Step-by-Step Solution

Answer

The maximum product is 64.

Step 1: Define the Variables

Let the two numbers be denoted as \(x\) and \(y\). According to the problem, these two numbers add up to 16, so \(x+y=16\). Using subtraction, the second number \(y\) can be defined as \(y=16-x\).

Step 2: Express the Product in Terms of One Variable

The product of these two numbers is \(x*y\). Substitute \(y\) from the first step into this to get a equation for the product all in terms of \(x\): \(xy=x(16-x)=16x-x²\).

Step 3: Find the Derivative of the Product

The derivative of the product function with respect to \(x\) is given by \(\frac{d(16x-x²)}{dx} = 16-2x\).

Step 4: Find the Maximum Product

Set the derivative equal to zero and solve for \(x\) to find the value that gives the maximum product: \(16-2x=0 \Rightarrow x=8\). Substitute \(x=8\) into the equation defining \(y\) from Step 1 gives \(y=16-8=8\). So, both numbers are \(8\).

Step 5: Calculate the Maximum Product

Substitute \(x=8\) and \(y=8\) into the product equation from Step 2 to get the maximum possible product: \(maxProduct = 8*8 = 64\).