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

`n=40`

Step-by-step 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`