the profit (p), in dollars, for a company is modeled by the function p(x) = -750x^2 + 15,000x, where x is the number of items produced. For which values of x will the company lose money?
your wrong
For the company to lose money, p(x)<0
We have: p(x)=x(15000-750x)
And since x is the number of items produced ==> x>0
Thus, 15000-750x<0;
750x>15000;
x>20
Therefore, the company loses money for x>20 items
The range of f(x) = ax + b is the set of all real numbers given that a and b are real numbers. Which produces a counter example to her statement?
To determine when the company will lose money, we need to find the values of x where the profit (p) is negative. In other words, we want to find the solutions to the inequality:
p(x) < 0
Given the profit function p(x) = -750x^2 + 15,000x, we substitute these values into the inequality:
-750x^2 + 15,000x < 0
Now, let's solve this inequality step by step:
Step 1: Factor out common terms:
x(-750x + 15,000) < 0
Step 2: Determine the critical points where the inequality might change:
Setting each factor equal to zero, we find:
x = 0 and -750x + 15,000 = 0
From the second equation, we solve for x:
-750x = -15,000
x = -15,000 / -750
x = 20
Therefore, we have the critical points:
x = 0 and x = 20
Step 3: Create a sign chart or number line. Mark the critical points:
---------------|---|---|-----------------------
0 20
Step 4: Test a value in each interval to determine the sign:
For x < 0 (the interval to the left of 0):
Choose x = -1 (a random value less than 0)
Substitute into the inequality:
-750(-1)^2 + 15,000(-1) < 0
750 - 15,000 < 0
-14,250 < 0
The sign is negative.
For 0 < x < 20 (the interval between 0 and 20):
Choose x = 10 (a random value between 0 and 20)
Substitute into the inequality:
-750(10)^2 + 15,000(10) < 0
-75,000 + 150,000 < 0
75,000 < 0
The sign is positive.
For x > 20 (the interval to the right of 20):
Choose x = 25 (a random value greater than 20)
Substitute into the inequality:
-750(25)^2 + 15,000(25) < 0
-468,750 + 375,000 < 0
-93,750 < 0
The sign is negative.
Step 5: Analyze the sign chart or number line:
From the sign chart, we see that the inequality is negative for x < 0 and x > 20.
Therefore, the company will lose money if the number of items produced (x) is less than 0 or greater than 20.