If the ratio of the radius to the height of a cylinder is $7 : 6$ and the curved surface area of the cylinder is 1056 $cm^2$, find the total surface area of the cylinder.

Question

If the ratio of the radius to the height of a cylinder is $7 : 6$ and the curved surface area of the cylinder is 1056  $cm^2$, find the total surface area of the cylinder.

Accepted Answer

**Solution:**

1. Let the radius of the cylinder be $7x$ and the height be $6x$ as per the given ratio $7:6$.

2. The curved surface area (CSA) of a cylinder is given by:
   $$	ext{CSA} = 2 \pi r h$$
   Substituting the values:
   $$2 	imes \frac{22}{7} 	imes 7x 	imes 6x = 1056$$

3. Simplify the equation:
   $$2 	imes \frac{22}{7} 	imes 7x 	imes 6x = 1056$$
   $$2 	imes 22 	imes 6 x^2 = 1056$$
   $$264 x^2 = 1056$$

4. Solve for $x^2$:
   $$x^2 = \frac{1056}{264} = 4$$
   $$x = 2 	ext{ cm}$$

5. Calculate the radius and height:
   - Radius $r = 7x = 7 	imes 2 = 14 	ext{ cm}$
   - Height $h = 6x = 6 	imes 2 = 12 	ext{ cm}$

6. The total surface area (TSA) of a cylinder is:
   $$	ext{TSA} = 2 \pi r (r + h)$$
   Substitute the values:
   $$2 	imes \frac{22}{7} 	imes 14 	imes (14 + 12)$$
   $$= 2 	imes \frac{22}{7} 	imes 14 	imes 26$$

7. Simplify step-by-step:
   $$= 2 	imes 22 	imes 2 	imes 26 = 2288 	ext{ cm}^2$$

**Answer:** $2288 	ext{ cm}^2$

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