The cost, in dollars, to produce \( x \) designer dog leashes is \( C(x)=9 x+6 \), and the price-demand function, in dollars per leash, is \( p(x)=105-4 x \) Find the profit function. \( P(x)= \) Find the number of leashes which need to be sold to maximize the profit. Select an answer

To find the profit function, we use the formula \( P(x) = R(x) - C(x) \), where \( R(x) \) is the revenue function and \( C(x) \) is the cost function. The revenue \( R(x) \) is given by \( R(x) = p(x) \cdot x = (105 - 4x)x = 105x - 4x^2 \). Therefore, the profit function is: \[ P(x) = R(x) - C(x) = (105x - 4x^2) - (9x + 6) = -4x^2 + 96x - 6. \] To maximize profit, we find the vertex of this quadratic function. The x-coordinate of the vertex is given by \( x = -\frac{b}{2a} \), where \( a = -4 \) and \( b = 96 \): \[ x = -\frac{96}{2 \cdot -4} = 12. \] Thus, the number of leashes which need to be sold to maximize the profit is 12.