Let's break down the problem step by step: --- ### **Step 1: Understanding the Fence** - **Base:** The base lies on the *upper half of the unit circle*, i.e., \( x^2 + y^2 \leq 1 \), and \( y \geq 0 \). - **Height:** At each point \((x, y)\), the height is \( f(x, y) = 2 - y \). --- ### **Step 2: Surface Area Calculation** Since the function for height is \( z = 2 - y \), and the surface is a vertical "curtain" over the upper half of the unit circle, the surface is **not tilted** (because \( z \) depends only on \( y \)), so we use the "graph surface area" formula: \[ \text{Area} = \iint_{D} \sqrt{ \left( \frac{\partial f}{\partial x} \right)^2 + \left( \frac{\partial f}{\partial y} \right)^2 + 1} \, dA \] For \( f(x, y) = 2 - y \): \[ \frac{\partial f}{\partial x} = 0, \qquad \frac{\partial f}{\partial y} = -1 \] So, \[ \sqrt{ (0)^2 + (-1)^2 + 1 } = \sqrt{2} \] Thus, \[ \text{Area} = \sqrt{2} \iint_{D} dA = \sqrt{2} \cdot (\text{Area of upper half unit circle}) \] --- #### **Area of the Upper Half Unit Circle** The area of the whole unit circle is \( \pi \cdot 1^2 = \pi \). The **upper half** is \( \frac{\pi}{2} \). --- #### **Total Surface Area** \[ \text{Total Area} = \sqrt{2} \cdot \frac{\pi}{2} \] --- ### **Step 3: Total Charge** - **Charge per square meter:** \( 2.00 \) pounds - **Total charge:** \( 2.00 \times \text{Total Area} \) \[ \text{Total Charge} = 2.00 \times \sqrt{2} \cdot \frac{\pi}{2} = \sqrt{2} \pi \approx 4.4429 \text{ pounds} \] --- ### **Step 4: Sketching the Fence** - The base is the upper semicircle \( x^2 + y^2 \leq 1 \), \( y \geq 0 \). - The height at each point is \( 2 - y \), so it decreases linearly from 2 at \( y=0 \) to 1 at \( y=1 \). --- ### **Step 5: Final Answers** #### **a) Surface area:** \[ \boxed{ \sqrt{2} \cdot \frac{\pi}{2} \text{ m}^2 \approx 2.221 \text{ m}^2 } \] #### **b) Total charge:** \[ \boxed{ \sqrt{2}\pi \text{ pounds} \approx 4.44 \text{ pounds} } \] --- **Explanation Summary:** - Used the formula for surface area of a graph. - Evaluated the area over the upper half unit circle. - Multiplied by the cost per unit area. If you need the explicit sketch, let me know!