The ratio of diameter and height of a right circular cylinder is $4 : 3$. If the diameter of the cylinder is reduced by 25\%, then its total surface area is reduced to $318.5\pi$ square meter. What is the circumference of the base of the cylinder?

Question

Accepted Answer

**Solution:**

1. Let the diameter be $d = 4x$ and height be $h = 3x$.

2. Radius $r = \frac{d}{2} = 2x$.

3. After reducing diameter by 25\%, new diameter = $0.75 	imes 4x = 3x$.
   New radius $r' = \frac{3x}{2} = 1.5x$.

4. Total surface area (TSA) of a right circular cylinder is given by:
   $$	ext{TSA} = 2\pi r (r + h)$$

5. Original TSA:
   $$2\pi 	imes 2x (2x + 3x) = 2\pi 	imes 2x 	imes 5x = 20\pi x^2$$

6. New TSA after diameter reduction:
   $$2\pi 	imes 1.5x (1.5x + 3x) = 2\pi 	imes 1.5x 	imes 4.5x = 13.5\pi x^2$$

7. The reduction in TSA is:
   $$20\pi x^2 - 13.5\pi x^2 = 6.5\pi x^2$$

8. Given the reduced TSA is $318.5\pi$, so:
   $$13.5\pi x^2 = 318.5\pi \implies 13.5 x^2 = 318.5$$

However, the problem states the reduced TSA is $318.5\pi$, so the equation is:
   $$2\pi \left[ 2x(2x + 3x) - \frac{3x}{2} \left( \frac{3x}{2} + 3x ight) ight] = 318.5\pi$$

Simplifying inside the bracket:
   $$2\pi \left[ 10x^2 - 6.75x^2 ight] = 318.5\pi$$
   $$2\pi 	imes 3.25x^2 = 318.5\pi$$
   $$6.5x^2 = 318.5$$
   $$x^2 = 49$$
   $$x = 7$$

9. Radius of original cylinder:
   $$r = 2x = 2 	imes 7 = 14$$

10. Circumference of base:
    $$2\pi r = 2\pi 	imes 14 = 28\pi$$

**Answer:** $28\pi 	ext{ cm}$

Answer

Answer

Answer