1. **State the problem:** We have a uniform density curve representing accidents along a 5-mile bike path with constant height $\frac{1}{5}$. We need to answer questions about the properties of this density curve and calculate proportions of accidents in certain intervals. 2. **Density curve requirements:** A density curve must satisfy two conditions: - It is on or above the horizontal axis (nonnegative everywhere). - The total area under the curve equals 1. 3. **Check the curve:** The curve is constant at height $\frac{1}{5}$ from 0 to 5 miles. - It is nonnegative since $\frac{1}{5} > 0$. - The area under the curve is height $\times$ base = $\frac{1}{5} \times 5 = 1$. 4. **Answer (a):** The correct reason is: "Because it is on or above the horizontal axis and the area beneath the curve is 1." 5. **Calculate area for (b):** Proportion of accidents in first 3 miles is area under curve from 0 to 3. $$\text{Area} = \text{height} \times \text{width} = \frac{1}{5} \times 3 = \frac{3}{5} = 0.6$$ 6. **Calculate area for (c):** Proportion of accidents between 0.2 and 0.4 miles: $$\text{Area} = \frac{1}{5} \times (0.4 - 0.2) = \frac{1}{5} \times 0.2 = 0.04$$ 7. **Calculate area for (d):** Accidents more than 1 mile from either road means between 1 mile and 4 miles (since total length is 5 miles). $$\text{Area} = \frac{1}{5} \times (4 - 1) = \frac{1}{5} \times 3 = 0.6$$ **Final answers:** - (a) Because it is on or above the horizontal axis and the area beneath the curve is 1. - (b) 0.6 - (c) 0.04 - (d) 0.6