The height of a right circular cylinder is 4 cm and its total surface area is 8π sq.cm. What is the radius (in cm) of the base of the cylinder?

Introduction / Context:
This problem requires using the total surface area formula for a right circular cylinder and solving a quadratic equation to find the radius. The height and total surface area are given, and you must express the radius in a simplified surd form.

Given Data / Assumptions:
• Height of the cylinder h = 4 cm.
• Total surface area of the cylinder S = 8π sq.cm.
• The cylinder is closed at both ends (includes both circular bases in surface area).
• Radius of the base is r cm, which we must find.

Concept / Approach:
For a closed cylinder, total surface area is given by S = 2πrh + 2πr2. Substitute the known values for S and h and solve the resulting quadratic equation in r. Then select the positive root, since radius must be positive and physically meaningful.

Step-by-Step Solution:
Step 1: Start with S = 2πrh + 2πr2. Step 2: Substitute S = 8π and h = 4: 8π = 2πr × 4 + 2πr2. Step 3: Simplify the right side: 2πr × 4 = 8πr, so 8π = 8πr + 2πr2. Step 4: Divide both sides by 2π to reduce coefficients: 8π / (2π) = 4, and 8πr / (2π) = 4r, and 2πr2 / (2π) = r2. This yields 4 = 4r + r2. Step 5: Rearrange to standard quadratic form: r2 + 4r − 4 = 0. Step 6: Solve the quadratic using the formula r = [−b ± √(b2 − 4ac)] / (2a), with a = 1, b = 4, c = −4. Step 7: Discriminant D = b2 − 4ac = 16 − 4 × 1 × (−4) = 16 + 16 = 32. Step 8: So r = [−4 ± √32] / 2 = [−4 ± 4√2] / 2 = −2 ± 2√2. Step 9: Only the positive root is valid, so r = −2 + 2√2 = 2√2 − 2 cm.

Verification / Alternative check:
Approximate √2 ≈ 1.414, so r ≈ 2 × 1.414 − 2 ≈ 2.828 − 2 = 0.828 cm. Substitute back: curved area 2πrh ≈ 2π × 0.828 × 4 ≈ 6.624π; base areas 2πr2 ≈ 2π × 0.685 ≈ 1.37π; total ≈ (6.624 + 1.37)π ≈ 7.994π, which is very close to 8π, confirming correctness with minor rounding.

Why Other Options Are Wrong:
• (2 − √2) cm and √2 cm do not satisfy the quadratic equation when substituted.
• 2 cm is far too large and would give a surface area significantly greater than 8π.

Common Pitfalls:
Mistakes often occur when rearranging the equation or handling the quadratic formula, especially with signs. Another pitfall is discarding the surd form too soon and over rounding, which can lead to inaccurate verification. Keeping expressions exact until the last step is safer.

Final Answer:
The radius of the base of the cylinder is (2√2 − 2) cm.