3D Geometry
Planes
Grade 12
Question:
<p>Distance between the planes is \(h = 5/\sqrt{6}\). The figure formed is a cylinder whose radius is \(r = 2\) units. Which of the following are correct?</p>
<p>(a) Distance between planes is \(\dfrac{5}{\sqrt{6}}\)</p>
<p>(b) Volume of cylinder is \(\dfrac{20\pi}{\sqrt{6}}\) cubic units</p>
<p>(c) Curved surface area is \(\dfrac{20\pi}{\sqrt{6}}\)</p>
<p>(d) Radius is \(r = 3\) units</p>
Step-by-Step Solution
Key Concept: A cylinder's volume and lateral surface area depend only on radius r and height h; recognizing that given h and r uniquely determines all geometric properties of the cylinder allows verification of multiple statements simultaneously.
Given: Cylinder with radius r = 2 and height h = 5/√6 Step 1: Volume of cylinder
V = πr^2h = π(2)^2(5/√6) = π(4)(5/√6) = 20π/√6 = (20π√6)/6 = (10π√6)/3 ✓ Step 2: Lateral Surface Area
LSA = 2πrh = 2π(2)(5/√6) = 20π/√6 = (10π√6)/3 ✓ Step 3: Total Surface Area
TSA = 2πrh + 2πr^2 = (10π√6)/3 + 2π(4) = (10π√6)/3 + 8π = π[(10√6)/3 + 8] = π[(10√6 + 24)/3] ✓ Step 4: Verification
All three standard formulas applied correctly with given values yield determinate expressions. Check which options match these calculated values. ∴ Answer: A, B, C (All standard cylinder formulas verified correctly)
Correct Answer: A, B, C