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Chapter 3: Problem 61

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

Short Answer

Expert verified
The maximum product is 64.

Step by step solution

01

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 \(y=16-x\).
02

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 \(x\): \(xy=x(16-x)=16x-x虏\).
03

Find the Derivative of the Product

The derivative of the product function with respect to \(x\) is given by \(\frac{d(16x-x虏)}{dx} = 16-2x\).
04

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 \(8\).
05

Calculate the Maximum Product

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

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