Answer:
A. To compute the breakeven units, we need to find the point where the total revenue equals the total cost, or R(x) = C(x). Substituting equations (1) and (2), we get:
(-x^2/100) + 10x = 700 + 2x
Simplifying and rearranging, we get:
-x^2/100 + 8x - 700 = 0
Multiplying both sides by -100 to eliminate the fraction, we get:
x^2 - 800x + 70000 = 0
Using the quadratic formula, we get:
x = (800 ± sqrt(800^2 - 4(1)(70000))) / 2(1)
x = (800 ± 400) / 2
x = 600 or 200
Therefore, the breakeven units are either 200 or 600.
If the company produces 200 units, it will break even but will not maximize profits. If it produces 600 units, it will also break even but will be operating at its maximum profit point. The company could choose to produce more than 600 units, but it would then incur losses due to the decreasing market price.
B. To illustrate the analysis using a graphic technique, we can plot the total revenue and total cost curves on the same graph.
First, we can solve for the total revenue and total cost at different levels of output, such as 0, 100, 200, 300, 400, 500, 600, 700, and 800 units, and then plot the points on a graph. Alternatively, we can use the equations (1) and (2) to generate the curves directly.
The total revenue curve is given by equation (1), which is a quadratic equation with a downward slope. The total cost curve is given by equation (2), which is a straight line with a positive slope. The breakeven point is where the two curves intersect.
Here is a graph that illustrates the CVP analysis:
As we can see from the graph, the breakeven point occurs at x = 200 and x = 600, where the total revenue curve intersects with the total cost curve. The point (200, 900) represents a breakeven point, but it is not optimal for profit maximization. The point (600, 1900) represents the optimal production level for profit maximization, where the company can break even and maximize profits.