1. **Problem 14:** Find the value of $k$ if the slope of the line through points $(3, 2)$ and $(8, k)$ is $\frac{3}{5}$. 2. **Formula for slope:** The slope $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ 3. **Apply the formula:** Using points $(3, 2)$ and $(8, k)$, the slope is $$\frac{k - 2}{8 - 3} = \frac{k - 2}{5}$$ 4. **Set equal to given slope:** $$\frac{k - 2}{5} = \frac{3}{5}$$ 5. **Multiply both sides by 5:** $$\cancel{5} \times \frac{k - 2}{\cancel{5}} = \cancel{5} \times \frac{3}{5}$$ $$k - 2 = 3$$ 6. **Solve for $k$:** $$k = 3 + 2 = 5$$ --- 7. **Problem 16:** Given line $\ell$ through $(-2, 0)$ and $(4, 3)$, and line $m$ through $(1, -1)$ and $(k, 1)$, find: (i) slope of $\ell$ (ii) slope of $m$ in terms of $k$ (iii) value of $k$ if $\ell \parallel m$ 8. **Slope of $\ell$:** $$m_\ell = \frac{3 - 0}{4 - (-2)} = \frac{3}{6} = \frac{1}{2}$$ 9. **Slope of $m$:** $$m_m = \frac{1 - (-1)}{k - 1} = \frac{2}{k - 1}$$ 10. **If $\ell \parallel m$, slopes are equal:** $$\frac{1}{2} = \frac{2}{k - 1}$$ 11. **Cross multiply:** $$\frac{1}{2} = \frac{2}{k - 1} \Rightarrow 1 \times (k - 1) = 2 \times 2$$ $$k - 1 = 4$$ 12. **Solve for $k$:** $$k = 4 + 1 = 5$$ --- 13. **Problem 17:** Check which ramps have gradient $\leq 0.08$. Given gradients: - A: 0.07 - B: 0.10 - C: 0.0625 - D: 0.0833 14. **Suitable ramps:** Those with gradient $\leq 0.08$ are A and C. **Final answers:** 14. $k = 5$ 16. (i) slope of $\ell = \frac{1}{2}$ (ii) slope of $m = \frac{2}{k - 1}$ (iii) $k = 5$ 17. Suitable ramps are A and C.