The marketing department estimates that if the selling price of the new product is set at birr 40 per unit sells will be 400 units per week if the selling price is birr 20 per unit sells will be 800 units per week the production department stimates that variable costs will be 7,50 per unit and fixed costs birr 10,000 per week then a devlope slope b devlope sales, cost and profit equation c find break even point?
A) Develop the slope:
First, we need to find the slope of the demand equation. We have two points: (40, 400) and (20, 800).
Slope = (change in y) / (change in x)
Slope = (800-400) / (20-40)
Slope = 400 / -20
Slope = -20
So, the slope of the demand equation is -20.
B) Develop sales, cost, and profit equations:
Sales (Q) = Slope × Price (P) + Initial sales (where slope = -20, and initial sales are found by putting Price=40)
Q = -20 × P + 800
Cost equation:
Total Cost (TC) = Variable Cost (VC) * Quantity (Q) + Fixed Cost (FC)
Variable Cost: 7.5 per unit
Fixed Cost: 10,000 per week
TC = 7.5*Q + 10,000
Revenue equation:
Total Revenue (TR) = Price * Quantity
TR = P * Q
Profit equation:
Total Profit (TP) = Total Revenue - Total Cost
TP = TR - TC
C) Find the break-even point:
At the break-even point, the total profit is zero (0):
0 = TR - TC
We need to find the Price and Quantity at which this occurs.
From the revenue equation, we know TR = P*Q. We can substitute Q from the demand equation:
TR = P*(-20*P + 800)
Then substitute TR in the profit equation:
0 = P*(-20*P + 800) - (7.5*(-20*P+800) + 10,000)
Now, solve for P:
0 = -20 * P^2 + 800 * P - 7.5 * (-20*P + 800) - 10,000
0 = -20 * P^2 + 800 * P + 150 * P - 6000
0 = -20 * P^2 + 950 * P - 6000
We can use the quadratic formula to solve for P:
P = [-b ± √(b²-4ac)] / 2a
Plugging in the values:
a = -20
b = 950
c = -6000
P = [-950 ± √(950^2-4*(-20)*(-6000))] / (2*(-20))
Solve this equation to find the break-even price:
P ≈ 24.74 (subtracting the square root part gives a negative price, which is not feasible)
Now, we find the break-even quantity using the demand equation:
Q = -20 * (24.74) + 800
Q ≈ 307.15
So, the break-even point is when the selling price is approximately 24.74 birr and the quantity sold is approximately 307 units per week.
To answer your questions, let's break it down step by step:
a) Develop the sales, cost, and profit equation:
The sales equation is calculated by multiplying the selling price per unit by the number of units sold per week. In this case:
If the selling price per unit is birr 40 and the number of units sold per week is 400:
Sales = Birr 40 * 400 units = Birr 16,000 per week
If the selling price per unit is birr 20 and the number of units sold per week is 800:
Sales = Birr 20 * 800 units = Birr 16,000 per week
The cost equation includes variable costs and fixed costs. Variable costs are calculated by multiplying the variable cost per unit by the number of units sold:
Variable Costs = Birr 7.50 * Number of Units Sold
Fixed costs are a constant value regardless of the number of units sold:
Fixed Costs = Birr 10,000 per week
As for the profit equation, it is calculated by subtracting the total costs from the total sales:
Profit = Sales - Total Costs
Now that we have the sales, cost, and profit equations, we can move on to the next question.
b) To develop the sales, cost, and profit equations, we have already outlined the process in the previous step.
c) Finding the break-even point:
The break-even point is the point at which the total revenue equals the total costs, resulting in zero profit. To calculate the break-even point, we need to set the profit equation equal to zero:
Profit = 0
Sales - Total Costs = 0
Sales = Total Costs
Substituting the values:
Birr 16,000 = Birr 7.50 * Number of Units Sold + Birr 10,000
Rearranging the equation:
Birr 7.50 * Number of Units Sold = Birr 16,000 - Birr 10,000
Birr 7.50 * Number of Units Sold = Birr 6,000
Finally, solving for the number of units sold:
Number of Units Sold = Birr 6,000 / Birr 7.50
Number of Units Sold = 800 units
Therefore, the break-even point is 800 units.
a) To develop the sales equation, we can use the given information:
At a selling price of birr 40 per unit, the number of units sold per week is 400.
At a selling price of birr 20 per unit, the number of units sold per week is 800.
Let's denote the selling price as SP and the number of units sold per week as Q. We can assume a linear relationship between SP and Q.
Using the two data points (40, 400) and (20, 800), we can calculate the slope (m) and the y-intercept (b) of the equation using the formula:
m = (Q2 - Q1) / (SP2 - SP1)
b = Q1 - m * SP1
Using (40, 400) and (20, 800):
m = (800 - 400) / (20 - 40) = 400 / (-20) = -20
b = 400 - (-20) * 40 = 400 + 800 = 1200
Therefore, the sales equation is:
Q = -20SP + 1200
b) To calculate the cost equation, we need the information about the variable costs and fixed costs.
Variable costs per unit: birr 7.50
Fixed costs per week: birr 10,000
The cost equation can be established using the formula:
Cost = (Variable cost per unit * Number of units) + Fixed costs
Cost = 7.50Q + 10,000
c) The break-even point is the level of sales where the total revenue equals total costs, resulting in zero profit. In equation form, the break-even point can be found by setting the profit equation equal to zero and solving for Q:
Total Revenue - Total Costs = 0
SP * Q - (7.50Q + 10,000) = 0
Using the sales equation Q = -20SP + 1200, we can substitute Q in the above equation:
SP * (-20SP + 1200) - (7.50(-20SP + 1200) + 10,000) = 0
Simplifying the above equation will give us the break-even point.