Questions: 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. Using this equation, find out the maximum amount of profit the company can make, to the nearest dollar. y = -5x^2 + 263x - 1844

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

y = -5x^2 + 263x - 1844

Transcript text: 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. Using this equation, find out the maximum amount of profit the company can make, to the nearest dollar. \[ y=-5 x^{2}+263 x-1844 \]

Solution

Solution Steps

Step 1: Calculate the selling price for maximum profit

To find the selling price that maximizes profit, we use the formula $x_{max} = -\frac{b}{2a}$. Substituting the given values, we get $x_{max} = -\frac{263}{2(-5)} = 26$.

Step 2: Calculate the maximum profit

Next, we calculate the maximum profit by substituting $x_{max}$ back into the profit function: $y_{max} = a(x_{max})^2 + b(x_{max}) + c$. This gives us $y_{max} = -5(26)^2 + 263(26) - 1844 = 1614$.