Understanding the Sphere Surface Area Problem

The problem asks us to find the final radius of a sphere after it has been enlarged. We know that the enlargement caused the radius to increase by 2 cm, and this resulted in the surface area increasing by a specific amount, 704 cm\(^2\). We are also given the value of pi to use in our calculations: \(\pi = \frac{22}{7}\).

Let's denote the original radius of the sphere as '\(r\)' cm. According to the problem, the radius is increased by 2 cm. So, the new radius will be '\(r + 2\)' cm.

Surface Area Formula and Equation Setup

The formula for the surface area '\(A\)' of a sphere with radius '\(R\)' is given by:

In our case, we have two scenarios:

• Original Surface Area (\(A_{old}\)) with radius '\(r\)': \(A_{old} = 4 \pi r^2\)
• New Surface Area (\(A_{new}\)) with radius '\(r+2\)': \(A_{new} = 4 \pi (r+2)^2\)

The problem states that the surface area increases by 704 cm\(^2\). This means:

Substituting the formulas:

Step-by-Step Calculation

Let's solve the equation step-by-step:

1. Factor out common terms: \(4 \pi [(r+2)^2 - r^2] = 704\)
2. Substitute the value of \(\pi\): \(4 \times \frac{22}{7} [(r+2)^2 - r^2] = 704\) \(\frac{88}{7} [(r+2)^2 - r^2] = 704\)
3. Expand the squared term \((r+2)^2\): \((r+2)^2 = r^2 + 2(r)(2) + 2^2 = r^2 + 4r + 4\). \(\frac{88}{7} [ (r^2 + 4r + 4) - r^2 ] = 704\)
4. Simplify the expression inside the brackets: The '\(r^2\)' terms cancel out. \(\frac{88}{7} [ 4r + 4 ] = 704\)
5. Isolate the term with '\(r\)': Multiply both sides by \(\frac{7}{88}\) to isolate '\(4r + 4\)'. \(4r + 4 = 704 \times \frac{7}{88}\) Since \(704 \div 88 = 8\), \(4r + 4 = 8 \times 7\) \(4r + 4 = 56\)
6. Solve for the original radius '\(r\)': Subtract 4 from both sides. \(4r = 56 - 4\) \(4r = 52\) Divide by 4: \(r = \frac{52}{4}\) \(r = 13 \text{ cm}\) This is the original radius.
7. Find the new radius: The question asks for the radius *after* the increase. New Radius = Original Radius + 2 cm New Radius = \(r + 2\) New Radius = \(13 + 2\) New Radius = \(15 \text{ cm}\)

Final Answer Determination

The calculation shows that the original radius was 13 cm. After increasing the radius by 2 cm, the new radius becomes 15 cm. This matches one of the options provided.