A manufacturer determines that the profit in dollars for manufacturing n units is P=2n^2-60x-400. (Assume that n is a positive integer) How many units are produced when the profit is $400?
400=2n^2-60n-400
solve the quadratic for n.
I will be happy to critique your thinking. Hint: n cannot be negative.
To find the number of units produced when the profit is $400, we need to set the profit equation equal to 400 and solve for n. The given profit equation is P = 2n^2 - 60n - 400.
Setting P equal to 400, we have:
400 = 2n^2 - 60n - 400
Adding 400 to both sides to isolate the terms containing n, we get:
2n^2 - 60n = 800
Now, let's rearrange the equation to make it easier to solve:
2n^2 - 60n - 800 = 0
To solve this equation, we can either factor it or use the quadratic formula. Since the equation cannot be easily factored, let's apply the quadratic formula:
n = (-b ± √(b^2 - 4ac)) / (2a)
Here, a = 2, b = -60, and c = -800. Substituting these values into the formula, we have:
n = (-(-60) ± √((-60)^2 - 4(2)(-800))) / (2(2))
Simplifying further:
n = (60 ± √(3600 + 6400)) / 4
n = (60 ± √10000) / 4
n = (60 ± 100) / 4
Now, we have two possible solutions:
1. n = (60 + 100) / 4 = 160 / 4 = 40
2. n = (60 - 100) / 4 = -40 / 4 = -10
Since the problem states that n is a positive integer, we discard the negative solution. Hence, when the profit is $400, the manufacturer produces 40 units.