Question

The profit, P, in dollars for manufacturing n units of a certain product is given by the formula P = 3n2 - 90n - 720. (Assume that n is a positive integer.) How many units are produced when the profit is $480.00?

Answer 1

Answer:

$n=40$

Explanation:

We want to know the value of $n$ when $P=480$

$P=3n^2-90n-720\\480=3n^2-90n-720\\3n^2-90n-720-480=0\\3n^2-90n-1200=0\\3(n^2-30n-400)=0\\n^2-30n-400=0$

From here, we can find the factors of the quadratic equation, we need two numbers that multiplied give -400 and added -30. Since they are factors of 400, we can choose -20x20 or -40x10. When adding -20 and 20 the result is zero, but the sum of -40 and 10 is -30. Then:

$n^2-30n-400=0\\(n-40)(n+10)=0$

The solutions of the quadratic equation are$n-40=0$ and $n+10=0$:

$n_1=40\\n_2=-10$

Since $n$ is a positive integer:

$n=40$