Download Maximize the product of two numbers

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.
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