Question

The surface area of a sphere is \(616 \text{ cm}^2\), its diameter (in cm) is :

• A. 14
• B. 7
• C. 21
• D. 10.5

Hint:

1. Surface area of a sphere: \(A = 4\pi r^2\). 2. Given \(A = 616 \text{ cm}^2\). Use \(\pi = 22/7\). \(616 = 4 \times \frac{22}{7} \times r^2\) \(616 = \frac{88}{7} r^2\) 3. Solve for \(r^2\): \(r^2 = \frac{616 \times 7}{88}\). Simplify: \(616/88 = (88 \times 7)/88 = 7\). So, \(r^2 = 7 \times 7 = 49\). 4. Find radius \(r\): \(r = \sqrt{49} = 7\) cm. 5. Find diameter \(d\): \(d = 2r = 2 \times 7 = 14\) cm.

Answer 1

The Correct Option is A

Concept: The surface area (\(A\)) of a sphere with radius \(r\) is given by the formula \(A = 4\pi r^2\). The diameter (\(d\)) of a sphere is twice its radius (\(d = 2r\)). Step 1: Given information and formula Given: Surface area of the sphere, \(A = 616 \text{ cm}^2\). Formula: \(A = 4\pi r^2\). We will use the approximation \(\pi \approx \frac{22}{7}\). Step 2: Substitute the given surface area into the formula to find the radius (\(r\)) \[ 616 = 4 \pi r^2 \] \[ 616 = 4 \times \frac{22}{7} \times r^2 \] \[ 616 = \frac{88}{7} \times r^2 \] Step 3: Solve for \(r^2\) To isolate \(r^2\), multiply both sides by \(\frac{7}{88}\): \[ r^2 = 616 \times \frac{7}{88} \] Let's simplify this. We can see if 616 is divisible by 88. \(616 \div 88\): Try dividing by common factors. 88 is \(8 \times 11\). Is 616 divisible by 11? \(6+6 - 1 = 11\), so yes. \(616 / 11 = 56\). So, \(r^2 = 56 \times \frac{7}{8}\). Now, \(56 / 8 = 7\). \[ r^2 = 7 \times 7 \] \[ r^2 = 49 \] Step 4: Solve for \(r\) Take the square root of both sides: \[ r = \sqrt{49} \] \[ r = 7 \text{ cm} \] (Since radius must be positive, we take the positive square root). Step 5: Calculate the diameter (\(d\)) The diameter \(d = 2r\). \[ d = 2 \times 7 \text{ cm} \] \[ d = 14 \text{ cm} \] The diameter of the sphere is \(14 \text{ cm}\). This matches option (1).