1. Using marginal cot functon \( C^{\prime}(q)=100+10 x-0.03 x^{2} \) for Producing genian, find the total cost in naira of producing the next 50 units for 100 units have been produced.

To determine the total cost of producing the next 50 units after already producing 100 units, we'll use the given **marginal cost function**: \[ C'(q) = 100 + 10q - 0.03q^2 \] Here, \( C'(q) \) represents the marginal cost when producing the \( q \)-th unit. ### Step 1: Understand the Problem - **Initial Production**: 100 units - **Additional Production**: 50 units (from 100 to 150 units) - **Total Additional Cost**: We need to find the cost of producing units 101 through 150. ### Step 2: Integrate the Marginal Cost Function The total additional cost \( \Delta C \) can be found by integrating the marginal cost function from \( q = 100 \) to \( q = 150 \): \[ \Delta C = \int_{100}^{150} (100 + 10q - 0.03q^2) \, dq \] ### Step 3: Compute the Integral Integrate term by term: \[ \begin{align*} \Delta C &= \int_{100}^{150} 100 \, dq + \int_{100}^{150} 10q \, dq - \int_{100}^{150} 0.03q^2 \, dq \\ &= 100q \bigg|_{100}^{150} + 5q^2 \bigg|_{100}^{150} - 0.01q^3 \bigg|_{100}^{150} \end{align*} \] Now, evaluate each term at the bounds: 1. **First Term**: \[ 100q \bigg|_{100}^{150} = 100(150) - 100(100) = 15,000 - 10,000 = 5,000 \] 2. **Second Term**: \[ 5q^2 \bigg|_{100}^{150} = 5(150)^2 - 5(100)^2 = 5(22,500) - 5(10,000) = 112,500 - 50,000 = 62,500 \] 3. **Third Term**: \[ -0.01q^3 \bigg|_{100}^{150} = -0.01(150)^3 + 0.01(100)^3 = -0.01(3,375,000) + 0.01(1,000,000) = -33,750 + 10,000 = -23,750 \] ### Step 4: Sum the Results \[ \Delta C = 5,000 + 62,500 - 23,750 = 43,750 \text{ Naira} \] ### Conclusion The **total additional cost** of producing the next 50 units (from 100 to 150 units) is **₦43,750**. **Answer:** The total cost of producing the next 50 units is ₦43 750.