a. The cost for HSMS can be given by the equation \( \mathrm{C}=20 \mathrm{~h}+30 \) where C is the cost in dollars for a given time and h is the time in hours. Using Desmos or any graphing software, draw the graph for HSMS, for \( t \) between 0 and 5 hours.

Certainly! Let's graph the cost function for HSMS using the equation: \[ \mathrm{C} = 20\,\mathrm{h} + 30 \] where: - \( \mathrm{C} \) is the cost in dollars, - \( \mathrm{h} \) is the time in hours. We'll graph this equation for \( \mathrm{h} \) ranging from 0 to 5 hours using **Desmos**, a free and user-friendly online graphing calculator. --- ## Step-by-Step Guide to Plotting the Graph in Desmos ### 1. **Access Desmos** - Open your web browser and navigate to the [Desmos Graphing Calculator](https://www.desmos.com/calculator). ### 2. **Input the Equation** - In the first input field, type the equation exactly as it is: ``` C = 20h + 30 ```  ### 3. **Restrict the Domain (0 ≤ h ≤ 5)** To ensure that the graph only shows the range where \( \mathrm{h} \) is between 0 and 5 hours, you'll need to add domain restrictions. - Modify the equation to include curly braces `{}` specifying the domain: ``` C = 20h + 30 {0 ≤ h ≤ 5} ```  ### 4. **Adjust the Viewing Window (Optional)** Desmos typically auto-adjusts the viewing window, but to ensure clarity: - **X-Axis (h):** Set the minimum to 0 and the maximum to 5. - **Y-Axis (C):** Given \( C = 20(5) + 30 = 130 \), you might set the y-axis from 0 to 150. To adjust, click on the wrench (graph settings) icon and input your desired ranges.  ### 5. **Plot Key Points (Optional)** For better understanding, you can plot specific points where \( h = 0, 1, 2, 3, 4, 5 \). | h (hours) | C (dollars) | |-----------|-------------| | 0 | 30 | | 1 | 50 | | 2 | 70 | | 3 | 90 | | 4 | 110 | | 5 | 130 | - To plot these points, enter them as separate expressions: ``` (0, 30) (1, 50) (2, 70) (3, 90) (4, 110) (5, 130) ```  ### 6. **Interpreting the Graph** - **Slope (\( 20 \)):** Represents the variable cost per hour. - **Y-intercept (\( 30 \)):** Represents the fixed cost when \( h = 0 \). The line will start at \( (0, 30) \) and increase linearly, reaching \( (5, 130) \) at 5 hours. ### 7. **Customize the Graph (Optional)** - **Style the Line:** Click on the colored icon next to the equation to change the line color, style, or add labels. - **Add Labels:** To label your axes, write "h (hours)" for the x-axis and "C (dollars)" for the y-axis. --- ## Visual Representation Here's how the final graph should approximately look:  *(Note: This is a representative image. When you follow the steps in Desmos, your graph will dynamically update.)* --- ## Alternative Graphing Software If you prefer using different graphing tools, the process is similar: - **GeoGebra:** [GeoGebra Graphing Calculator](https://www.geogebra.org/graphing) - **Microsoft Excel:** Use the "Insert Scatter" feature and add a trendline. - **Google Sheets:** Similar to Excel, utilize the chart tools to plot the equation. --- ## Summary By following the steps above, you can visualize how the cost \( \mathrm{C} \) increases linearly with time \( \mathrm{h} \) for HSMS within the specified range of 0 to 5 hours. This graph helps in understanding the cost structure, distinguishing between fixed and variable costs. If you have any further questions or need additional assistance with graphing or interpreting the equation, feel free to ask!