A solid cylinder and a solid cone have equal base and equal height. If the radius and height be in the ratio of $4 : 3$, the ratio of the total surface area of the cylinder to that of the cone is

### Concept & Total Surface Area (TSA) Proportion To find the ratio of their total surface areas, we need the standard TSA formulas. Since the shapes share the same radius ($r$) and height ($h$), we use their given ratio to find the cone's slant height ($l$). $$TSA_{cylinder} = 2\pi r(r + h)$$ $$TSA_{cone} = \pi r(r + l)$$ $$l = \sqrt{r^2 + h^2}$$ ### Step-by-Step Solution * **Step 1:** Establish proportional values. Let radius $r = 4x$ and height $h = 3x$. * **Step 2:** Calculate the slant height of the cone. $l = \sqrt{(4x)^2 + (3x)^2} = \sqrt{16x^2 + 9x^2} = \sqrt{25x^2} = 5x$ * **Step 3:** Calculate the Total Surface Area of the cylinder. $TSA_{cylinder} = 2\pi (4x)(4x + 3x) = 8\pi x (7x) = 56\pi x^2$ * **Step 4:** Calculate the Total Surface Area of the cone. $TSA_{cone} = \pi (4x)(4x + 5x) = 4\pi x (9x) = 36\pi x^2$ * **Step 5:** Determine the ratio. $Ratio = \frac{56\pi x^2}{36\pi x^2} = \frac{56}{36}$ Divide both numerator and denominator by 4: $Ratio = \frac{14}{9}$ ### Exam Strategy & Shortcut Recognize the classic $3-4-5$ right triangle. If $r=4$ and $h=3$, then $l=5$. Set up the ratio directly: $\frac{2(r + h)}{r + l}$ (since $\pi r$ cancels out). Substitute the values: $\frac{2(4 + 3)}{4 + 5} = \frac{2(7)}{9} = \frac{14}{9}$. This takes 10 seconds. ### Common Pitfall A common mistake is using the Curved Surface Area ($2\pi rh$ and $\pi rl$) instead of Total Surface Area, which would give a ratio of $\frac{24}{20} = \frac{6}{5}$, leading to an incorrect answer. Always verify if the question asks for "curved" or "total" surface area. ### Final Answer Therefore, the correct answer is **14 : 9**.