1. **Problem 1a:** Find the angle measure in radians when the bug travels 5.2 feet around the circumference of a circle with radius 2.6 feet. 2. **Formula:** The arc length $s$ of a circle is related to the radius $r$ and the angle in radians $\theta$ by the formula: $$s = r \theta$$ 3. **Step 1:** Rearrange the formula to solve for $\theta$: $$\theta = \frac{s}{r}$$ 4. **Step 2:** Substitute $s = 5.2$ feet and $r = 2.6$ feet: $$\theta = \frac{5.2}{2.6}$$ 5. **Step 3:** Simplify the fraction: $$\theta = \frac{\cancel{5.2}}{\cancel{2.6} \times 2} = 2$$ 6. **Answer 1a:** The angle swept out is $2$ radians. --- 1. **Problem 1b:** Find the angle measure in radians when the bug travels 11.7 feet around the circumference. 2. **Step 1:** Use the formula $\theta = \frac{s}{r}$. 3. **Step 2:** Substitute $s = 11.7$ feet and $r = 2.6$ feet: $$\theta = \frac{11.7}{2.6}$$ 4. **Step 3:** Simplify the fraction: $$\theta = \frac{\cancel{11.7}}{\cancel{2.6} \times 4.5} = 4.5$$ 5. **Answer 1b:** The angle swept out is $4.5$ radians. --- 1. **Problem 1c:** Find the angle measure in radians when the bug travels $d$ feet. 2. **Step 1:** Use the formula $\theta = \frac{s}{r}$. 3. **Step 2:** Substitute $s = d$ and $r = 2.6$ feet: $$\theta = \frac{d}{2.6}$$ 4. **Answer 1c:** The angle swept out is $\frac{d}{2.6}$ radians. --- 1. **Problem 1d:** If the angle measure remains the same and the radius doubles, how will the arc length change? 2. **Step 1:** Original arc length is $s = r \theta$. 3. **Step 2:** New radius is $2r$, so new arc length is: $$s_{new} = 2r \theta = 2s$$ 4. **Answer 1d:** The arc length doubles. --- 1. **Problem 2a:** Find the arc length when the angle swept out is 1.5 radians. 2. **Step 1:** Use $s = r \theta$. 3. **Step 2:** Substitute $r = 2.6$ feet and $\theta = 1.5$ radians: $$s = 2.6 \times 1.5 = 3.9$$ 4. **Answer 2a:** The bug travels 3.9 feet. --- 1. **Problem 2b:** Find the arc length when the angle swept out is 4.7 radians. 2. **Step 1:** Use $s = r \theta$. 3. **Step 2:** Substitute $r = 2.6$ feet and $\theta = 4.7$ radians: $$s = 2.6 \times 4.7 = 12.22$$ 4. **Answer 2b:** The bug travels 12.22 feet. --- 1. **Problem 3:** Choose three different angle measures (in radians) and estimate the bug's vertical distance above the horizontal diameter. 2. **Step 1:** The vertical distance $y$ is given by: $$y = r \sin(\theta)$$ 3. **Step 2:** Choose angles $\theta = 1, 2, 3$ radians. 4. **Step 3:** Calculate vertical distances: $$y_1 = 2.6 \sin(1) \approx 2.6 \times 0.8415 = 2.19$$ $$y_2 = 2.6 \sin(2) \approx 2.6 \times 0.9093 = 2.36$$ $$y_3 = 2.6 \sin(3) \approx 2.6 \times 0.1411 = 0.37$$ 5. **Answer 3:** The points to plot are approximately $(1, 2.19)$, $(2, 2.36)$, and $(3, 0.37)$ on the graph.