XYZ Company plans to market a new product.Based on its market studies,the company estimates that it can sell 5500 units in 2004.The selling price will be birr 2 per unit.Variable costs are estimated to be 40% of the selling price.Fixed costs are estimated to be $ 600 A)Develop the revenue,cost and profit functions interns of sales and quantity B) What is the break even point in units and in $? C) If the company faces a loss of 4000 $ what will be the sales? D)Support your answer using the break-even chart (Show all the necessary lines and points E) If the company is sure of selling 5500 units every year,Determine the least price that should be charged in order to guarantee no loss.
From the first sentence, you know that if x units are produced,
cost = 600 + 0.4x
income = 2x
Why 2x? You say they sell for birr 2 each. Yet you note all other amounts in terms of $, so I assume that 1 birr = 1 $. So I assume a birr is not some small fraction of a $. If that is wrong, then nake the necessary changes. And in the future, use the same currency terms throughout, or do not assume that everyone knows what the relationships are. If I did a similar problem using florins and farthings, would you know what to do?
Now, knowing that profit p(x) = income-cost,
B) find x when p(x) = 0
C) find the change in x when p = -4000
E) solve p(x) >= 0
A) To develop the revenue, cost, and profit functions, let's define the variables:
Q = quantity of units sold
P = selling price per unit
Revenue (R) = Q * P
Cost (C) = Variable cost per unit * Q + Fixed costs
Profit (Pf) = Revenue - Cost
Given the information from the question, the selling price per unit is $2, and the variable costs are 40% of the selling price, so Variable cost per unit = 0.4 * $2 = $0.8. The fixed costs are $600.
Revenue Function (R) = Q * P = Q * $2
Cost Function (C) = Variable cost per unit * Q + Fixed costs = $0.8 * Q + $600
Profit Function (Pf) = R - C = Q * $2 - ($0.8 * Q + $600)
B) The break-even point is the quantity at which the company neither makes a profit nor incurs a loss. At the break-even point, the profit is zero. We can set the profit function equal to zero and solve for the break-even point:
Pf = 0
Q * $2 - ($0.8 * Q + $600) = 0
Q * $2 - $0.8Q - $600 = 0
$2Q - $0.8Q = $600
$1.2Q = $600
Q = $600 / $1.2
Q = 500 units
To find the break-even point in dollars, multiply the break-even quantity by the selling price:
Break-even point in dollars = 500 units * $2/unit = $1000
C) If the company faces a loss of $4000, we can again set the profit function equal to the loss amount and solve for the sales:
Pf = -$4000
Q * $2 - ($0.8 * Q + $600) = -$4000
Solve this equation for Q to find the sales quantity.
D) To support the answers using a break-even chart, we would need to plot the revenue, cost, and profit lines on a graph. The horizontal axis would represent the quantity of units sold (Q), and the vertical axis would represent the amount in dollars.
1. Draw the revenue line: Start from the origin (0,0) and draw a line that passes through the point (Quantity, Revenue).
2. Draw the cost line: Start from the origin (0,0) and draw a line that passes through the point (Quantity, Cost).
3. Draw the profit line: Start from the origin (0,0) and draw a line that passes through the point (Quantity, Profit).
The break-even point would be the point at which the profit line intersects the x-axis (quantity axis).
E) If the company is sure of selling 5500 units every year, we can determine the least price that should be charged to guarantee no loss. We can set up the profit function Pf = R - C and solve for the selling price P:
Pf = 0 (to ensure no loss)
5500P - (0.4 * 5500P + $600) = 0
Solve this equation for P to find the least price that should be charged.