Questions: The total profit P(x) in thousands of dollars from the sale of x hundred thousand aluminium tires is approximated by P(x) = -x^2 + 156x + 724 - 400. x ≥ 5. Find the maximum profit.

Solution Steps

To find the maximum profit, we need to determine the vertex of the quadratic function P(x) = -x^2 + 156x + 324. The vertex of a parabola given by a quadratic equation in the form of ax^2 + bx + c is at x = -b/(2a). This x-value will give us the number of hundred thousand tires that need to be sold to achieve the maximum profit. We then substitute this x-value back into the profit function to find the maximum profit.

The profit function is given by

\[ P(x) = -x^2 + 156x + 324 \]

To find the maximum profit, we calculate the vertex of the parabola. The x-coordinate of the vertex is given by

\[ x = -\frac{b}{2a} \]

Substituting \(a = -1\) and \(b = 156\):

\[ x = -\frac{156}{2 \cdot -1} = 78.0 \]

Next, we substitute \(x = 78.0\) back into the profit function to find the maximum profit:

\[ P(78) = - (78)^2 + 156 \cdot 78 + 324 \]

Calculating this gives:

\[ P(78) = -6084 + 12168 + 324 = 6408.0 \]

Final Answer