When the radius of a sphere is increased by 5 cm, its surface area increases by `704 cm^2`. The diameter of the original sphere, is (Take `pi = (22)/(7))`

Options

A. 8.2 cm
B. 6.8 cm
C. 5.2 cm
D. 6.2 cm

Answer

To solve the problem step by step, let's follow the given information and apply the formulas for the surface area of a sphere.

### Step 1: Define the original radius
Let the original radius of the sphere be \( r \) cm.

### Step 2: Define the new radius
When the radius is increased by 5 cm, the new radius becomes:
\[ r + 5 \text{ cm} \]

### Step 3: Write the formula for the surface area
The surface area \( A \) of a sphere is given by the formula:
\[ A = 4\pi r^2 \]
Thus, the original surface area \( A_1 \) is:
\[ A_1 = 4\pi r^2 \]
And the new surface area \( A_2 \) with the new radius is:
\[ A_2 = 4\pi (r + 5)^2 \]

### Step 4: Calculate the increase in surface area
According to the problem, the increase in surface area is given as:
\[ A_2 - A_1 = 704 \text{ cm}^2 \]
Substituting the expressions for \( A_1 \) and \( A_2 \):
\[ 4\pi (r + 5)^2 - 4\pi r^2 = 704 \]

### Step 5: Factor out common terms
Factoring out \( 4\pi \):
\[ 4\pi \left((r + 5)^2 - r^2\right) = 704 \]

### Step 6: Simplify the expression
Now, simplify the expression inside the parentheses:
\[ (r + 5)^2 - r^2 = (r^2 + 10r + 25) - r^2 = 10r + 25 \]
Thus, we have:
\[ 4\pi (10r + 25) = 704 \]

### Step 7: Substitute \( \pi \)
Using \( \pi = \frac{22}{7} \):
\[ 4 \cdot \frac{22}{7} (10r + 25) = 704 \]

### Step 8: Clear the fraction
Multiply both sides by 7 to eliminate the fraction:
\[ 4 \cdot 22 (10r + 25) = 704 \cdot 7 \]
Calculating \( 704 \cdot 7 \):
\[ 704 \cdot 7 = 4928 \]
Thus, we have:
\[ 88 (10r + 25) = 4928 \]

### Step 9: Divide by 88
Now, divide both sides by 88:
\[ 10r + 25 = \frac{4928}{88} \]
Calculating \( \frac{4928}{88} \):
\[ \frac{4928}{88} = 56 \]
So, we have:
\[ 10r + 25 = 56 \]

### Step 10: Solve for \( r \)
Subtract 25 from both sides:
\[ 10r = 56 - 25 \]
\[ 10r = 31 \]
Now, divide by 10:
\[ r = \frac{31}{10} = 3.1 \text{ cm} \]

### Step 11: Calculate the diameter
The diameter \( D \) of the original sphere is given by:
\[ D = 2r = 2 \times 3.1 = 6.2 \text{ cm} \]

### Final Answer
The diameter of the original sphere is \( 6.2 \text{ cm} \).

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