\text{Expected Profit} = \frac{\int_{0.1}^{19} \left[ \min(5x, 7D - 2x) \right] \, dD}{19 - 0.1}
Since there's a uniform distribution, we consider the two cases where the profit equations exist:
1. **Demand \( D \geq x \) :**
\text{Profit} = 5x
2. **Demand \( D < x \) :**
\text{Profit} = 7D - 2x
Integrate over both segments accordingly:
E[\text{Profit}] = \int_{0.1}^{x} \left( 7D - 2x \right) \frac{dD}{18.9} + \int_{x}^{19} 5x \frac{dD}{18.9}
Solve these integrals:
E[\text{Profit}] = \left( \frac{7}{18.9} \int_{0.1}^{x} D \, dD - \frac{2x}{18.9} \int_{0.1}^{x} dD \right) + \frac{5x}{18.9} \int_{x}^{19} dD
= \frac{7}{18.9} \cdot \left[ \frac{D^2}{2} \right]_{0.1}^{x} - \frac{2x}{18.9} \left[ D \right]_{0.1}^{x} + \frac{5x}{18.9} \left[ D \right]_{x}^{19}
Evaluating these:
1. First part:
= \frac{7}{18.9} \left( \frac{x^2}{2} - \frac{0.1^2}{2} \right) - \frac{2x}{18.9} (x - 0.1)
2. Second part:
= \frac{5x}{18.9} (19 - x)
Combine and simplify:
E[\text{Profit}] = \frac{7x^2 - 7 \cdot 0.01}{2 \times 18.9} - \frac{2x^2 - 0.2x}{18.9} + \frac{5x (19 - x)}{18.9}
After simplifying, the expected profit equation in terms of \( x \):
E[\text{Profit}] = \frac{(7x^2 - 0.07) - (2x^2 - 0.2x) + 5x(19 - x)}{18.9}
Set the derivative of accumulated profit with respect to \( x \) to 0 and solve for \( x \):
\frac{d}{dx} \left[ \frac{5x (19 - x) + 7 \frac{x^2}{2} - 0.07 - 2x^2 + 0.2x}{18.9} \right] = 0
\frac{d}{dx} \left[ \frac{13.6x - \frac{7x^2}{2} - 0.07 - 2x^2 + 0.2x}{18.9} \right] =0
13.6 \quad \text{results in maximum expected profit}
Therefore:
x = 13.6