Question

A company sells widgets . The amount of profit, Y, made by the company , is related to the selling price of each widget, X, by the given equation :- y = - 12x² + 718x - 5419

Using this equation, find out the maximum amount of profit the company can make, to the nearest dollar.

Answer 1

The maximum amount of profit the company can make is approximately $9371.

Step-by-step explanation:

To find the maximum amount of profit, we need to determine the vertex of the quadratic equation y = -12x² + 718x - 5419. The vertex of a quadratic equation in the form ax² + bx + cis given by the formula
`\(x = -(b)/(2a)\)`. In this case, a = -12 and b = 718.

`\[x = -(718)/(2 * -12) = (718)/(24) = 29.92\]`

Now that we have the value of x, we can substitute it back into the original equation to find the maximum profit (y).

`\[y = -12 * (29.92)^2 + 718 * 29.92 - 5419\]`

After calculating this expression, we get
`\(y \approx 9370.67\)`. Rounding to the nearest dollar, the maximum amount of profit the company can make is approximately $9371.

This result makes sense because the negative coefficient of the x² term indicates a downward-opening parabola, and the calculated maximum profit corresponds to the vertex of this parabola. Therefore, $9371 represents the peak of the profit function, and any other values of (x) would result in a lower profit.