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A television manufacturer makes rear-projection and plasma televisions. The profit per unit is $\$ 125$ for the rear-projection televisions and $\$ 200$ for the pla...

Question

A television manufacturer makes rear-projection and plasma televisions. The profit per unit is $\$ 125$ for the rear-projection televisions and $\$ 200$ for the plasma televisions. a. Let $x=$ the number of rear-projection televisions manufactured in a month and $y=$ the number of plasma televisions manufactured in a month. Write the objective function that models the total monthly profit. b. The manufacturer is bound by the following constraints: $\bullet$ Equipment in the factory allows for ma

A television manufacturer makes rear-projection and plasma televisions. The profit per unit is $\$ 125$ for the rear-projection televisions and $\$ 200$ for the plasma televisions. a. Let $x=$ the number of rear-projection televisions manufactured in a month and $y=$ the number of plasma televisions manufactured in a month. Write the objective function that models the total monthly profit. b. The manufacturer is bound by the following constraints: $\bullet$ Equipment in the factory allows for making at most 450 rear-projection televisions in one month. $\bullet$ Equipment in the factory allows for making at most 200 plasma televisions in one month. $\bullet$ The cost to the manufacturer per unit is $\$ 600$ for the rear-projection televisions and $\$ 900$ for the plasma televisions. Total monthly costs cannot exceed $\$ 360,000$. Write a system of three inequalities that models these constraints. c. Graph the system of inequalities in part (b). Use only the first quadrant and its boundary, because $x$ and $y$ must both be nonnegative. d. Evaluate the objective function for total monthly profit at each of the five vertices of the graphed region. [The vertices should occur at $(0,0),(0,200),(300,200),$ $(450,100),$ and $(450,0) .]$ e. Complete the missing portions of this statement: The television manufacturer will make the greatest profit by manufacturing ______ rear-projection televisions each month and _______ plasma televisions each month. The maximum monthly profit is \$ ______.



Answers

MAXIMUM PROFIT The cost per unit in the production of an MP3 player is $\$60$. The manufacturer charges $\$90$ per unit for orders of 100 or less. To encourage large orders,
the manufacturer reduces the charge by $\$0.15$ per MP3 player for each unit ordered in excess of 100 (for example, there would be a charge of $\$87$ per MP3 player for an order size of 120).

(a)
The table shows the profits $P$ (in dollars) for various numbers of units ordered, $x$. Use the table to estimate the maximum profit.

(b)
Plot the points $(x, P)$ from the table in part (a). Does the relation defined by the ordered pairs represent as a function of $x$?

(c)
If $P$ is a function of $x$, write the function and determine its domain.


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