Let’s now return to the problem posed at the beginning of the section.
Golf balls and lagrange multipliers
The golf ball manufacturer, Pro-T, has developed a profit model that depends on the number
of golf balls sold per month (measured in thousands), and the number of hours per month of advertising
y , according to the function
where
is measured in thousands of dollars. The budgetary constraint function relating the cost of the production of thousands golf balls and advertising units is given by
Find the values of
and
that maximize profit, and find the maximum profit.
Again, we follow the problem-solving strategy:
The optimization function is
To determine the constraint function, we first subtract 216 from both sides of the constraint, then divide both sides by
which gives
The constraint function is equal to the left-hand side, so
The problem asks us to solve for the maximum value of
subject to this constraint.
So, we calculate the gradients of both
The equation
becomes
which can be rewritten as
We then set the coefficients of
equal to each other:
The equation
becomes
Therefore, the system of equations that needs to be solved is
We use the left-hand side of the second equation to replace
in the first equation:
Then we substitute this into the third equation:
Since
this gives
We then substitute
into
which gives
Therefore the maximum profit that can be attained, subject to budgetary constraints, is
with a production level of
golf balls and
hours of advertising bought per month. Let’s check to make sure this truly is a maximum. The endpoints of the line that defines the constraint are
and
Let’s evaluate
at both of these points:
The second value represents a loss, since no golf balls are produced. Neither of these values exceed
so it seems that our extremum is a maximum value of
A company has determined that its production level is given by the Cobb-Douglas function
where
x represents the total number of labor hours in
year and
y represents the total capital input for the company. Suppose
unit of labor costs
and
unit of capital costs
Use the method of Lagrange multipliers to find the maximum value of
subject to a budgetary constraint of
per year.
A maximum production level of
occurs with
labor hours and
of total capital input.
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