A theater company's annual profit is modeled by the equation P= 0.2x^2-6.8x, where x is the number of plays performed in a year. How many plays must the company present in order to make a profit?
Solve P = 0 and take the positive of the two roots.
0.2x^2 -6.8x = 0
0.2x(x - 34) = 0
x = 34 plays produce break even.
35 productions (performances?) will produce a profit.
No wonder theaters go broke.
To determine the number of plays the theater company must present in order to make a profit, we need to find the value of x that corresponds to a positive profit (P > 0) in the given equation.
The profit equation is P = 0.2x^2 - 6.8x.
Setting P greater than 0, we get:
0.2x^2 - 6.8x > 0.
Now let's solve this inequality step by step:
Step 1: Factor out the common term x:
x(0.2x - 6.8) > 0.
Step 2: Set each factor equal to zero to find the critical values:
x = 0 (from x = 0)
0.2x - 6.8 = 0 (solving for x gives x = 34).
Step 3: Create an interval chart with the critical values:
|---------------------|-----------------------|------------------------|
-∞ 0 34 +∞
Step 4: Test a value within each interval to determine the sign of the inequality:
For x < 0, we can choose x = -1: (-1)(0.2(-1) - 6.8) = (-1)(-0.2 - 6.8) = (-1)(-7) = 7. This is positive (+).
For 0 < x < 34, we can choose x = 10: (10)(0.2(10) - 6.8) = (10)(2 - 6.8) = (10)(-4.8) = -48. This is negative (-).
For x > 34, we can choose x = 40: (40)(0.2(40) - 6.8) = (40)(8 - 6.8) = (40)(1.2) = 48. This is positive (+).
Step 5: Analyze the signs and determine the solution:
From the chart, we see that the inequality is satisfied when x < 0 and x > 34. However, the question asks for the number of plays needed to make a profit, so negative plays are not relevant.
Therefore, the theater company must present more than 34 plays in order to make a profit.