The volume of a cylinder is $5280 ext{ m}^3$ and its curved surface area is $704 ext{ m}^2$. Find the ratio of the radius to the height of the cylinder.

Question

The volume of a cylinder is $5280 		ext{ m}^3$ and its curved surface area is $704 		ext{ m}^2$. Find the ratio of the radius to the height of the cylinder.

Accepted Answer

**Solution:**

1. Let the radius be $r$ and the height be $h$.

2. Given:
   - Volume of cylinder, $V = \pi r^2 h = 5280$
   - Curved Surface Area (CSA), $= 2 \pi r h = 704$

3. From the CSA formula:
   $$2 \pi r h = 704 \implies r h = \frac{704}{2 \pi} = \frac{704}{2 		imes 3.14} = \frac{704}{6.28} = 112$$

4. From the volume formula:
   $$\pi r^2 h = 5280 \implies r^2 h = \frac{5280}{\pi} = \frac{5280}{3.14} = 1681$$

5. Using $r h = 112$, express $h = \frac{112}{r}$.

6. Substitute $h$ in $r^2 h = 1681$:
   $$r^2 		imes \frac{112}{r} = 1681 \implies 112 r = 1681 \implies r = \frac{1681}{112}$$

7. Calculate $r$:
   $$r = 15$$

8. Calculate $h$:
   $$h = \frac{112}{r} = \frac{112}{15} = 7.4667$$

9. Ratio of radius to height:
   $$r : h = 15 : 7.4667 = 225 : 112$$

**Answer:** $225 : 112$

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