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OpenStax-CNX module: m48581 1 Maximize the product of two numbers ∗ John Taylor This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 3.0† Abstract Demonstrates nding the two numbers with the maximum product and also satisfying a linear equation. 1 Maximize the product of two numbers (The small steps below can be used for a self test. To do so, Scroll in small increments.) Two numbers are to be selected so that their product is a maximum. The sum of one number and 3 times the other number is 36. Determine the two numbers. To solve such a problem, we need variables for each number, say a and b. Express the rst condition in terms of these variables. a + 3 ∗ b = 36. Equation 1 Express the second condition in terms of P , the unknown product. The second condition is P = a ∗ b. Equation 2 How do we proceed? To maximize this, we need to express P in terms of one variable and dierentiate. To do that, we need to rst solve equation 1 for a. a = 36 − 3 ∗ b. Equation 3 Substitute this for a in Equation 2. P = (36 − 3 ∗ b) ∗ b = 36b − 3b2 Now we can dierentiate and set the derivative to zero to nd the maximum. First nd the derivative for any value of b: dP = 36 − 6 ∗ b db Then nd the maximum at bmax . dP | = 36 − 6 ∗ bmax = 0, or db bmax bmax = 6. This is one of our two numbers. Now we can use this in Equation 3 to nd the corresponding value of amax : amax = 36 − 3 ∗ bmax = 36 − 3 ∗ 6 = 18 So the two numbers are 18 and 6, with the maximum product of 18 ∗ 6 = 108. As a check, we can calculate the product for other pairs which satisfy equation 1, such as 21, 5 and 15, 7. ∗ Version 1.1: Jan 1, 2014 6:13 am -0600 † http://creativecommons.org/licenses/by/3.0/ http://cnx.org/content/m48581/1.1/ OpenStax-CNX module: m48581 2 In both cases the product in equation 2 is 105, smaller than the maximum of 108. If you found this helpful and would recommend that I create more pages like this one, please let me know using the email link at the top of the page. http://cnx.org/content/m48581/1.1/