Question

The net profit in dollars per day for a small business owner is given by the equation f(x) = -0.1x^2 + 6 x + 1, where x is the number of employees he hires. If he hires the number of employees that will maximize his profit, what will his profit be in dollars per day? (Enter an exact number.)

Answer 1

His profit in dollars per day is 91.

Explanation:

You know that the net profit in dollars per day for a small business owner is given by the equation f(x) = -0.1*x² + 6*x + 1, where x is the number of employees he hires.

If he hires the number of employees that will maximize his profit, then this indicates that I should look for the maximum of the function f (x).

A quadratic function or second degree function is a polynomial function defined as:

f(x)= a*x² + b*x + c

If the scalar a> 0, the parabola opens upwards and the vertex is the minimum of the function. On the other hand, if a <0, the parabola opens downwards and the vertex is the maximum of the function.

In this case, having a value of -0.1, the vertex will indicate the maximum of the function.

The maximum in x is then reached when:

`x=(-b)/(2*a)`

In this case, being a=-0.1 and b=6, you get:

`x=(-6)/(2*(-0.1))=(-6)/(-0.2) =30`

The number of employees who will maximize their profits is 30. So, replacing in the function f (x) you get:

f(30) = -0.1*30² + 6*30 + 1

Solving:

f(30)= 91

His profit in dollars per day is 91.