Question:
The volume of a right circular cylinder is \(554.4\text{ cm}^3\) and its height is 10 cm. What is the perimeter of the base of the cylinder? (Use \(\pi = \frac{22}{7}\))
एक लम्बवृत्तीय बेलन का आयतन \(554.4\text{ cm}^3\) है और इसकी ऊँचाई 10 cm है। बेलन के आधार का परिमाप क्या है? (\(\pi = \frac{22}{7}\) का उपयोग करें)
Options:
- 17.6 cm
- 26.4 cm
- 22 cm
- 44 cm
- 17.6 cm
- 26.4 cm
- 22 cm
- 44 cm
Correct Answer: b)
Explanation:
Step 1: Formula for the volume of a cylinder:
\[ V = \pi r^2 h \]
Given: \(V = 554.4\text{ cm}^3\), \(h = 10\text{ cm}\), and \(\pi = \frac{22}{7}\).
Step 2: Find the radius \(r\):
\[ \frac{22}{7} \times r^2 \times 10 = 554.4 \]
\[ \frac{22}{7} \times r^2 = 55.44 \]
\[ r^2 = \frac{55.44 \times 7}{22} = 2.52 \times 7 = 17.64 \]
\[ r = \sqrt{17.64} = 4.2\text{ cm} \]
Step 3: Calculate the perimeter of the circular base:
The base is a circle with circumference \(P = 2\pi r\):
\[ P = 2 \times \frac{22}{7} \times 4.2 = 2 \times 22 \times 0.6 = 44 \times 0.6 = 26.4\text{ cm} \]
Hence, the correct answer is option b).
चरण 1: बेलन के आयतन का सूत्र:
\[ V = \pi r^2 h \]
दिया गया है: \(V = 554.4\text{ cm}^3\), \(h = 10\text{ cm}\), तथा \(\pi = \frac{22}{7}\)।
चरण 2: त्रिज्या \(r\) ज्ञात करें:
\[ \frac{22}{7} \times r^2 \times 10 = 554.4 \]
\[ r^2 = \frac{55.44 \times 7}{22} = 2.52 \times 7 = 17.64 \]
\[ r = \sqrt{17.64} = 4.2\text{ cm} \]
चरण 3: आधार का परिमाप ज्ञात करें:
आधार एक वृत्त है जिसका परिमाप \(P = 2\pi r\) होता है:
\[ P = 2 \times \frac{22}{7} \times 4.2 = 44 \times 0.6 = 26.4\text{ cm} \]
अतः सही उत्तर विकल्प b) है।