Let's break down the solution step by step: ### **Step 1: Understand the Data** The table shows the number of pencils produced each year: | Year | Pencils Produced | |------|------------------| | 2010 | 25 | | 2011 | 37 | | 2012 | 52 | | 2013 | 74 | --- ### **Step 2: Graph the Data** Let's check which graph matches the data points: #### Plot the points (Year, Pencils Produced): - (2010, 25) - (2011, 37) - (2012, 52) - (2013, 74) Now, look at the provided graphs: - **Option A:** This graph increases rapidly, matching the data values. - **Option B:** This graph decreases, which doesn't match. - **Option C:** This graph is flat, doesn't match. - **Option D:** This graph decreases, doesn't match. **Correct graph: Option A.** --- ### **Step 3: Find the Average Rate of Increase** The average rate of increase from 2010 to 2013: \[ \text{Average Rate of Increase} = \frac{\text{Change in Number of Pencils}}{\text{Change in Years}} \] \[ \text{Change in Number of Pencils} = 74 - 25 = 49 \] \[ \text{Change in Years} = 2013 - 2010 = 3 \] \[ \text{Average Rate} = \frac{49}{3} \approx 16.33 \text{ pencils per year} \] --- ### **Step 4: Final Answers** **A. Draw the correct graph:** - **Option A** is correct. **B. The average rate of increase:** - \(\boxed{16.33}\) pencils per year (rounded to two decimal places). --- **Summary Table:** | Question | Answer | |----------|----------------------------| | A | Option A (correct graph) | | B | 16.33 pencils per year | Let me know if you need further explanation!

Certainly! Let's proceed with **Question C**, assuming it relates to the same data set. --- ### **Question C: Find the total number of pencils produced over the years 2010 to 2013.** --- ### **Step 1: Recall the data** | Year | Pencils Produced | |-------|------------------| | 2010 | 25 | | 2011 | 37 | | 2012 | 52 | | 2013 | 74 | --- ### **Step 2: Sum the production over the four years** Total pencils produced = sum of pencils each year: \[ 25 + 37 + 52 + 74 \] Calculate step-by-step: \[ 25 + 37 = 62 \] \[ 62 + 52 = 114 \] \[ 114 + 74 = 188 \] --- ### **Final Answer:** \[ \boxed{188} \text{ pencils} \] **The total number of pencils produced from 2010 to 2013 is 188.** --- If you need additional questions or clarifications, feel free to ask!

Let's interpret the question carefully: **"Use your graph to estimate the rate at which the pencils were changing in 2011."** --- ### **Step 1: Recall the data for 2011** From the table: | Year | Pencils Produced | |-------|------------------| | 2011 | 37 | --- ### **Step 2: Approximate change in production around 2011** Since the data points are for the years 2010 and 2012, we can estimate the rate of change in 2011 by considering the change between these years. - Pencils in 2010: 25 - Pencils in 2012: 52 --- ### **Step 3: Calculate the average rate of change between 2010 and 2012** \[ \text{Change in pencils} = 52 - 25 = 27 \] \[ \text{Change in years} = 2012 - 2010 = 2 \] \[ \text{Average rate} = \frac{27}{2} = 13.5 \text{ pencils per year} \] Since 2011 is exactly midway, the **rate of change in 2011** can be approximated as **13.5 pencils per year**. --- ### **Step 4: Using the graph** - On the graph, locate the point corresponding to 2011. - Observe the slope of the graph at 2011. - The slope (rise over run) gives the rate of change. Given the data and the approximate linear increase, the graph's slope around 2011 is close to **13.5 pencils per year**. --- ### **Final answer:** **The estimated rate at which pencils were changing in 2011 is approximately \(\boxed{13.5}\) pencils per year.** --- If you need a more precise estimate or further explanation, let me know!

Certainly! Here's a detailed step-by-step solution with calculations and explanations for each part based on the data and graph provided. --- ### Given Data: | Year | Pencils Produced | |-------|------------------| | 2010 | 25 | | 2011 | 37 | | 2012 | 52 | | 2013 | 74 | --- ### **Part A: Identifying the Correct Graph** **Step 1:** Plot the data points: - (2010, 25) - (2011, 37) - (2012, 52) - (2013, 74) **Step 2:** Examine the graphs (Options A, B, C, D). - Graph A shows a steady increasing trend consistent with the data points. - Graphs B and D show decreasing trends, which do not match. - Graph C is flat, also inconsistent. **Conclusion:** The correct graph is **Option A**. --- ### **Part B: Calculating the Average Rate of Increase (2010 to 2013)** **Step 1:** Find the change in the number of pencils: \[ \Delta \text{Pencils} = \text{Pencils in 2013} - \text{Pencils in 2010} = 74 - 25 = 49 \] **Step 2:** Find the change in years: \[ \Delta \text{Years} = 2013 - 2010 = 3 \] **Step 3:** Calculate the average rate of increase: \[ \text{Average Rate} = \frac{\Delta \text{Pencils}}{\Delta \text{Years}} = \frac{49}{3} \approx 16.33 \text{ pencils/year} \] **Final answer for Part B:** \[ \boxed{16.33 \text{ pencils per year}} \] --- ### **Part C: Total number of pencils produced from 2010 to 2013** **Step 1:** Sum the production for each year: \[ 25 + 37 + 52 + 74 \] **Step 2:** Calculate step-by-step: - \(25 + 37 = 62\) - \(62 + 52 = 114\) - \(114 + 74 = 188\) **Final answer for Part C:** \[ \boxed{188 \text{ pencils}} \] --- ### **Part D: Estimating the rate of change in 2011** Since we don't have a specific rate for just 2011, we approximate it using data from adjacent years. **Step 1:** Use data for 2010 and 2012: - Pencils in 2010: 25 - Pencils in 2012: 52 **Step 2:** Calculate the average rate over 2 years: \[ \frac{52 - 25}{2012 - 2010} = \frac{27}{2} = 13.5 \text{ pencils/year} \] **Step 3:** Since 2011 is midway, the rate in 2011 is approximately the same: \[ \boxed{13.5 \text{ pencils per year}} \] **Optional:** Using the graph, this slope can be visually confirmed to be around 13.5. --- ### **Summary of Final Answers:** | Part | Answer | |--------|---------| | A | **Option A (correct graph)** | | B | **16.33 pencils per year** (average rate from 2010 to 2013) | | C | **188 pencils** (total from 2010 to 2013) | | D | **Approximately 13.5 pencils per year** (rate in 2011) | --- ### **Note:** All calculations are based on the given data points and the assumption that the graph's shape matches the data trend. --- If you'd like further clarification or additional parts, feel free to ask!

Final Answers: **Part A:** Option A (correct graph) **Part B:** 16.33 pencils per year **Part C:** 188 pencils **Part D:** Approximately 13.5 pencils per year

Given Data: | Year | Number of Pencils Produced | |-------|----------------------------| | 2010 | 25 | | 2011 | 37 | | 2012 | 52 | | 2013 | 74 |

Given Data: - 2010: 25 pencils - 2011: 37 pencils - 2012: 52 pencils - 2013: 74 pencils