1. **Problem Statement:** If the height of a cylinder is increased by 15% and the radius of its base is reduced by 10%, find the percentage change in the surface area of the cylinder. 2. **Formula for Surface Area of Cylinder:** The total surface area $S$ of a cylinder with radius $r$ and height $h$ is given by: $$S = 2\pi r^2 + 2\pi r h$$ where $2\pi r^2$ is the area of the two circular bases and $2\pi r h$ is the lateral surface area. 3. **Changes in dimensions:** - New height $h' = h + 0.15h = 1.15h$ - New radius $r' = r - 0.10r = 0.90r$ 4. **New surface area $S'$:** $$S' = 2\pi (r')^2 + 2\pi r' h' = 2\pi (0.90r)^2 + 2\pi (0.90r)(1.15h)$$ $$= 2\pi (0.81r^2) + 2\pi (1.035 r h) = 2\pi r^2 (0.81) + 2\pi r h (1.035)$$ 5. **Original surface area $S$:** $$S = 2\pi r^2 + 2\pi r h$$ 6. **Calculate the ratio $\frac{S'}{S}$:** $$\frac{S'}{S} = \frac{2\pi r^2 (0.81) + 2\pi r h (1.035)}{2\pi r^2 + 2\pi r h} = \frac{0.81 (2\pi r^2) + 1.035 (2\pi r h)}{2\pi r^2 + 2\pi r h}$$ 7. **Let $x = \frac{h}{r}$ to simplify:** $$\frac{S'}{S} = \frac{0.81 (2\pi r^2) + 1.035 (2\pi r^2 x)}{2\pi r^2 + 2\pi r^2 x} = \frac{0.81 + 1.035 x}{1 + x}$$ 8. **Interpretation:** The percentage change depends on $x = \frac{h}{r}$. Without $x$, we cannot find exact percentage change. 9. **Assuming $h = r$ (i.e., $x=1$) for calculation:** $$\frac{S'}{S} = \frac{0.81 + 1.035}{1 + 1} = \frac{1.845}{2} = 0.9225$$ 10. **Percentage change:** $$\text{Change} = (0.9225 - 1) \times 100 = -7.75\%$$ This means surface area decreases by 7.75% if $h = r$. 11. **Check options:** Options are (a) 2.5% increase, (b) 3.5% increase, (c) 2.5% decrease, (d) 3.5% decrease. Since our calculated change is about 7.75% decrease, none exactly matches. 12. **Alternate approach:** If lateral surface area dominates (height much larger than radius), then: $$\frac{S'}{S} \approx \frac{1.035}{1} = 1.035$$ which is a 3.5% increase. If base area dominates (radius much larger than height), then: $$\frac{S'}{S} \approx 0.81$$ which is a 19% decrease. 13. **Conclusion:** The problem likely expects the lateral surface area to dominate, so the surface area increases by approximately 3.5%. **Final answer:** 3.5% increase (option b).