Answer: 95 units.
Step-by-step explanation:
The cost of making x units is:
C(x) = x^2 + 10*x + 45.
Now, if the cost is $10,020, then we can solve this for x as:
C(x) = 10,020 = x^2 + 10*x + 45
x^2 + 10*x + 45 - 10,020 = 0
x^2 + 10*x - 9,975 = 0
Now, remember that for a quadratic equation:
a*x^2 + b*x + c = 0
the solutions are:
![x = \frac{-b +-\sqrt{b^2 - 4*a*c} }{2*a}](https://tex.z-dn.net/?f=x%20%3D%20%5Cfrac%7B-b%20%2B-%5Csqrt%7Bb%5E2%20-%204%2Aa%2Ac%7D%20%7D%7B2%2Aa%7D)
In this case the solutions are:
x = ![x = \frac{-10 +-\sqrt{10^2 - 4*1*(-9,975)} }{2*1} = \frac{-10 +- 200}{2}](https://tex.z-dn.net/?f=x%20%3D%20%5Cfrac%7B-10%20%2B-%5Csqrt%7B10%5E2%20-%204%2A1%2A%28-9%2C975%29%7D%20%7D%7B2%2A1%7D%20%3D%20%5Cfrac%7B-10%20%2B-%20200%7D%7B2%7D)
Then we have two solutions, one for each sign:
x = (-10 -200)/2 = -110
x = (-10 + 200)/2 = 95
Here we must choose the positive option, as x represents a positive quaintity.
Then the number of units manufactured is 95.