SOLUTION: Could you please help me solve this word problem?
Congratulations! You have just been hired as a manager of Todaro’s, a small
business that makes frozen pizzas to sell at local
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-> SOLUTION: Could you please help me solve this word problem?
Congratulations! You have just been hired as a manager of Todaro’s, a small
business that makes frozen pizzas to sell at local
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Question 1075426: Could you please help me solve this word problem?
Congratulations! You have just been hired as a manager of Todaro’s, a small
business that makes frozen pizzas to sell at local markets. The owner gave
you the following information:
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Preparation and packaging takes 0.2 hours for each box of 12-inch pizzas
and 0.25 hours for each box of 16-inch pizzas.
-
You can pay the staff for no more than 240 hours of labor each week.
-
The staff must meet the company quota of producing at least 1,000 boxes
of pizza per week.
-
Todaro’s will make a profit of $2 for each box of 12-inch pizzas and $4
for each box of 16-inch pizzas. How many pizzas of each size must be
produced in order to maximize the profit? (15 pts.)
Let x represent the number of 12-inch pizzas made and let y represent
the number of 16-inch pizzas made.
a. Write an inequality for each constraint. (Hint: There are 4.)
b. Graph the feasible region that represents the constraints in
(a). (Note, you must submit a graph to earn full credit.)
c. Locate and label the vertices for the feasible region.
d. Write the objective function.
e. Find the number of each type of pizza made that will maximize
Todaro’s profit.
thanks! Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website! i'll provide the solution using the calculator at www.desmos.com.
this calculator allows you to graph inequalities and also allows you to present the equations as ax + by = c rather than y = (c-ax)/b.
the general procedure is:
determine the constraint functions.
graph the opposite of the inequalities.
the feasible region will be the area of the graph that is NOT shaded.
the corner points of the feasible region will contain the max / min solution.
you evaluate the objective function at those corner points and then choose the corner point that gives you the max/min solution.
you then check the constraint to see that they are all satisfied at that corner point.
you will see what i mean as we go through the problem.
your objective function is profit = 2x + 4y.
your constraint functions are:
x >= 0
y >= 0
.2x + .25y <= 240
x + y >= 1000
x + y >= 1000 means the number of pizza have to be greter than or equal to 1000.
.2x + .25y <= 240 means that the hours of labor to make the small pizza and the hours of labor to make the large pizza have to be less than or equal to 240.
x >= 0 and y >= 0 mean that the number of pizzas made can't be negative.
you will graph the opposite of the inequalities shown.
you will graph:
x <= 0
y <= 0
.2x + .25y >= 240
x + y <= 1000
each corner point is shown as (x,y), where x is the value of the x-coordinate of the point and y is the value of the y-coordinate of the point.
the corner points are at (200,800), (1000,0) (1200,0)
the value of the objective function when x = 200 and y = 800 will be:
profit = 2*200 + 4*800 = 400 + 3200 = 3600.
this will be your maximum profit.
you can calculate the objective function at each of the other corner points to see that this is true.
the value of your constraint functions when x = 200 and when y = 800 will be:
.2x + .25y = .2*200 + .25*400 = 240
x = 200
y = 800
x + y = 1000.
all the constraints are met.
x and y are >= 0
.2x + .25y is <= 240.
x + y is >= 1000.
your graph will look like this:
the first graph is the far out view.
the second graph is the near end view showing you the corner points of the feasible region.
