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.
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?
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