A company is planning to manufacture and sell a new headphone set. After conducting extensive market surveys, the research department provide the following estimates: Marginal costs function: c^' (x)=RM(40+0.4x) where x is the quantity sold for the fixed cost of RM 600. At the price of RM 100, 30 units of headphones were sold and at the price of RM 60, 50 units of headphones were sold
(ii) Find the total cost function (4 marks)
(iii) Find the total revenue function (2 marks)
(iv) The level of production in which revenue is maximised (3 marks)
(v) The profit, price per unit, total revenue and total cost when total revenue is maximised. (4 marks)
(vi) Find the change in total revenue when production level increases from 20 units to 30 units. Interpret your results. (4 marks)
(vii) Find the profit function (3 marks)
(viii) The level of production in which profit is maximised. (3 marks)
(ix) The maximum profit, price per unit, total revenue and total cost when profit is maximised (4 marks)
(x) Find the change in total profit when production level increases from 30 units to 40 units. Interpret your results. (4 marks)
(xi) Find the break-even of the production level
(ii) To find the total cost function, we need to integrate the marginal cost function c'(x).
c(x) = ∫(40 + 0.4x)dx
= 40x + 0.4 * (1/2 * x^2) + C
Given that the fixed cost is RM 600, we can find the constant C by substituting the values of x and c(x) for the given quantities.
When x = 30, c(30) = 600
600 = 40*30 + 0.4 * (1/2 * 30^2) + C
Simplifying the equation, we get:
600 = 1200 + 1800 + C
600 = C + 3000
C = -2400
Therefore, the total cost function is c(x) = 40x + 0.2x^2 - 2400.
(iii) The total revenue function can be calculated by multiplying the price (p) by the quantity sold (x).
At a price of RM 100, 30 units of headphones were sold:
Total Revenue at p1 = 100 * 30 = RM 3000
At a price of RM 60, 50 units of headphones were sold:
Total Revenue at p2 = 60 * 50 = RM 3000
Since the total revenue remained the same, we can conclude that the total revenue function is linear.
(iv) To find the level of production in which revenue is maximized, we need to find the maximum point of the revenue function. Since the total revenue function is linear and the slope is constant, the maximum revenue is attained when the highest possible quantity is sold.
In this case, the highest quantity sold is 50 units.
(v) When total revenue is maximized, the level of production is 50 units.
- Profit = Total Revenue - Total Cost = Total Revenue - Total Variable Cost - Fixed Cost
- Price per unit = Total Revenue / Quantity = Total Revenue / 50
- Total Revenue = Price per unit * Quantity
- Total Cost = Total Variable Cost + Fixed Cost
(vi) To find the change in total revenue when production level increases from 20 units to 30 units, we need to calculate the total revenue at both points.
At a price of RM 60, 20 units of headphones were sold:
Total Revenue at 20 units = 60 * 20 = RM 1200
At a price of RM 60, 30 units of headphones were sold:
Total Revenue at 30 units = 60 * 30 = RM 1800
Change in Total Revenue = Total Revenue at 30 units - Total Revenue at 20 units
= 1800 - 1200
= RM 600
Interpretation: The change in total revenue when production level increases from 20 units to 30 units is RM 600. This indicates that increasing production by 10 units results in an additional revenue of RM 600.
(vii) The profit function can be calculated by subtracting the total cost function from the total revenue function.
Profit = Total Revenue - Total Cost
= (Price per unit * Quantity) - (Variable Cost per unit * Quantity) - Fixed Cost
(viii) To find the level of production in which profit is maximized, we need to find the maximum point of the profit function.
(ix) When profit is maximized, the level of production is determined by finding the maximum point of the profit function. We can substitute the values of x from the given quantity of headphones sold and find the level that gives the maximum profit.
(x) To find the change in total profit when production level increases from 30 units to 40 units, we need to calculate the total profit at both points.
(xi) The break-even point of the production level is the quantity at which the total revenue equals the total cost, resulting in zero profit. This can be calculated by setting the profit function equal to zero and solving for x.
To answer these questions, we need to find the total cost function, total revenue function, profit function, and perform optimization to find the production level at which revenue and profit are maximized. Let's break down each step:
(i) Given data:
- Price at which 30 headphones were sold: RM 100
- Price at which 50 headphones were sold: RM 60
- Fixed cost: RM 600
(ii) To find the total cost function, we need to integrate the marginal cost function. The marginal cost function is given as c'(x) = RM(40 + 0.4x). By integrating it, we can find the total cost function:
Integral of c'(x) = Integral of RM(40 + 0.4x) dx
Total cost function, C(x) = Integral of RM(40 + 0.4x) dx
C(x) = RM(40x + 0.2x²) + C1, where C1 is the constant of integration
(iii) To find the total revenue function, we need to multiply the quantity sold by the price:
Total revenue function, R(x) = Price * Quantity sold
(iv) To find the production level at which revenue is maximized, we can use the information given for different price levels and quantities sold. We'll compare the total revenues for different scenarios and find the highest value.
(v) To find the profit when revenue is maximized, we need to subtract the total cost from the total revenue. The price per unit can be calculated by dividing the total revenue by the quantity sold.
(vi) To find the change in total revenue when production level increases from 20 units to 30 units, we can find the difference between the total revenues at these two levels. Then, we can interpret the result in terms of the impact on revenue.
(vii) To find the profit function, we need to subtract the total cost from the total revenue:
Profit function, P(x) = R(x) - C(x)
(viii) To find the production level at which profit is maximized, we can use optimization techniques such as setting the derivative of the profit function equal to zero and solving for x.
(ix) To find the maximum profit, price per unit, total revenue, and total cost when profit is maximized, we substitute the production level that maximizes profit into the relevant functions.
(x) To find the change in total profit when production level increases from 30 units to 40 units, we can find the differences between the total profits at these two levels. Then, we can interpret the result in terms of the impact on profit.
(xi) To find the break-even production level, we need to determine the production level at which the total revenue equals the total cost.