Bell Motors manufactures two models of electric cars. The...

Bell Motors manufactures two models of electric cars. The demand equations giving the relationship between the unit price, $p$ and $q$, and the number of cars demanded per year, $x$ and $y,$ of Models $S 1$ and S2 are $$ p=60,000-4 x-2 y $$ and $$ q=50,000-2 x-4 y $$ respectively. a. What is the total yearly revenue function $R(x, y) ?$ b. What is the domain of $R ?$ Sketch the domain $R$. c. Is the point (3000,2000) in the domain of $R ?$ Interpret your result. Hint: Show that $x=3000$ and $y=2000$ satisfy the system of inequalities obtained in part (b). d. What is the total revenue realized by Bell Motors if it sells 3000 Model S1s and 2000 Model S2s?

Bell Motors manufactures two models of electric cars. The demand equations giving the relationship between the
unit price, $p$ and $q$, and the number of cars demanded per year, $x$ and $y,$ of Models $S 1$ and S2 are
$$
p=60,000-4 x-2 y
$$
and
$$
q=50,000-2 x-4 y
$$
respectively.
a. What is the total yearly revenue function $R(x, y) ?$
b. What is the domain of $R ?$ Sketch the domain $R$.
c. Is the point (3000,2000) in the domain of $R ?$ Interpret your result.
Hint: Show that $x=3000$ and $y=2000$ satisfy the system of inequalities obtained in part (b).
d. What is the total revenue realized by Bell Motors if it sells 3000 Model S1s and
2000 Model S2s?

Key Concepts

Feasible Region in Economic Modeling

The feasible region is the collection of all points that satisfy a given system of constraints or inequalities. In revenue and optimization problems, this region represents all possible combinations of product quantities that yield valid or realistic outcomes, and it is frequently visualized graphically to assist in decision-making.

Revenue Function

A revenue function represents the total income generated from the sale of goods or services, calculated as the sum of each product’s price multiplied by the quantity sold. In many contexts, especially when prices depend on quantities (or vice versa), this function helps in understanding how changes in sales volume or pricing strategies impact overall earnings.

Demand Function

A demand function captures the relationship between the price of a product and the quantity that consumers are willing to purchase. Typically, these functions reflect the inverse relationship between price and demand, meaning that as the price increases, the quantity demanded tends to decrease, and vice versa.

Domain of a Function

The domain of a function refers to the set of all input values for which the function is defined. In economic models, this often includes natural constraints such as non-negative quantities and prices since negative values wouldn’t make sense in the context of sales and market demand.

System of Linear Inequalities

Systems of linear inequalities are used to define multiple simultaneous constraints on decision variables. In economic and business contexts, these systems often arise from conditions like non-negative prices and quantities, ensuring that the model remains realistic and applicable to real-world situations.

Verification of Feasible Solutions

Verifying whether a specific point lies within the domain or feasible region involves substituting the point into each inequality of the system. This process confirms that the proposed quantities or prices adhere to all the necessary constraints, ensuring that the solution is practically and economically viable.

Transcript

00:01 In this problem let us first look at part a total revenue function are x comma y will be given as the unit price for model s1 times the number of units sold of s1 which is x plus the unit price for s2 multiplied by the total number of units sold for s 2 which is y so this becomes 60 ,000 x minus 4x squared minus 2xy and this is plus 50 ,000 y minus 2xy minus 2xy this becomes minus 4x squared minus 4y squared minus 4y square minus 4y square minus 4xy plus 60 ,000 x plus 50 ,000 this is the revenue function let's look at part b of the problem which is the domain for the domain he has to be greater than equal to 0 x has to be greater than equal to 0 q has to be little than equal to 0 and y has to look greater than equal to so let's substitute we get 60 ,000 minus 4x minus 2y should be greater than equal to 0 from here we get 4x plus 2y should be less than equal to 60 ,000 or 2x plus y should be less than equals to 30 ,000 let's look at part b which is x greater than 0 and then q is 3 0 which means 50 ,000 minus 2x minus 4y should be greater than equal to 0 from here we get 2x plus 4y should be less than equals to 50 ,000 or x plus 2y should be less than equals to 25 ,000 next is y greater than equal to 0 so these are the domains let us plot this.

03:07 This is y.

03:08 This is x.

03:10 First, we need 2x plus y less than equals to 30 ,000.

03:15 So let's plot this.

03:18 This is the line.

03:20 This point is 15 ,000.

03:27 And this point is 30 ,000.

03:30 We need the region less than this.

03:34 So the region is towards this.

03:41 Next we have x greater than equal to 0 which means this line...