Question

The daily profit in dollars made by an automobile

manufacturer is

P(x) = -35x2 +2,100x - 20,000

where x is the number of cars produced per shift. Find the

maximum possible daily profit.

Answer 1

Answer:

The maximum possible daily profit is $11,500.

Step-by-step explanation:

Vertex of a quadratic function:

Suppose we have a quadratic function in the following format:

$f(x) = ax^{2} + bx + c$

It's vertex is the point $(x_{v}, f(x_{v})$

In which

$x_{v} = -\frac{b}{2a}$

If a<0, the vertex is a maximum point, that is, the maximum value happens at $x_{v}$, and it's value is $f(x_{v})$

In this question:

The maximum daily profit happens when $x_{v}$ cars are sold. This profit is $P(x_{v})$

$P(x) = -35x^{2} + 2100x - 20000$

So $a = -35, b = 2100$

$x_{v} = -\frac{2100}{2*(-35)} = 30$

The maximum possible daily profit is:

$P(30) = -35*30^2 + 2100*30 - 20000 = 11500$

The maximum possible daily profit is $11,500.