(c) The selling price $$ x $$ in Ksh. of each item is related to the total sales $$ y $$ by $$ T=400-0.5 x^{2} $$ The cost of production $$ c $$ of the item is related to the total sales by, $$ c=90+0.25 y $$. Determine the selling price that will result in the largest possible profit, giving your answer to one decimal place. Hence, determine the largest possible profit.

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To determine the selling price $$ x $$ that maximizes profit and the corresponding largest possible profit, we'll follow these steps:

### **1. Define the Given Relationships**

- **Total Sales ($$ y $$)**:
  $$
  y = 400 - 0.5x^2
  $$
  
- **Cost of Production ($$ c $$)**:
  $$
  c = 90 + 0.25y = 90 + 0.25(400 - 0.5x^2) = 90 + 100 - 0.125x^2 = 190 - 0.125x^2
  $$
  
- **Total Revenue ($$ R $$)**:
  $$
  R = x \cdot y = x(400 - 0.5x^2) = 400x - 0.5x^3
  $$
  
- **Profit ($$ P $$)**:
  $$
  P = R - c = (400x - 0.5x^3) - (190 - 0.125x^2) = -0.5x^3 + 0.125x^2 + 400x - 190
  $$

### **2. Find the Selling Price that Maximizes Profit**

To maximize profit, take the derivative of $$ P $$ with respect to $$ x $$ and set it to zero:

$$
\frac{dP}{dx} = -1.5x^2 + 0.25x + 400 = 0
$$

Solve the quadratic equation:
$$
-1.5x^2 + 0.25x + 400 = 0
$$
$$
6x^2 - x - 1600 = 0 \quad (\text{Multiplying both sides by } -4)
$$

Using the quadratic formula:
$$
x = \frac{1 \pm \sqrt{1 + 4 \cdot 6 \cdot 1600}}{12} = \frac{1 \pm \sqrt{38401}}{12}
$$

$$
\sqrt{38401} \approx 195.96
$$

$$
x = \frac{1 + 195.96}{12} \approx 16.4 \text{ Ksh}
$$

### **3. Calculate the Largest Possible Profit**

Substitute $$ x = 16.4 $$ into the profit equation:

$$
P = -0.5(16.4)^3 + 0.125(16.4)^2 + 400(16.4) - 190
$$
$$
P \approx -0.5(4410.944) + 0.125(268.96) + 6560 - 190
$$
$$
P \approx -2205.472 + 33.62 + 6560 - 190 \approx 4198.148 \text{ Ksh}
$$

### **Final Answer**

- **Optimal Selling Price**: $$ \mathbf{16.4} $$ Ksh
- **Largest Possible Profit**: $$ \mathbf{4198.1} $$ Ksh

### **Summary**

To achieve the maximum profit, set the selling price at **16.4 Ksh** per item, resulting in a largest possible profit of approximately **4198.1 Ksh**.
To maximize profit, set the selling price at **16.4 Ksh** per item, resulting in a largest possible profit of approximately **4198.1 Ksh**.

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