Two cylinders, A and B, are given. The radius of cylinder B is twice that of cylinder A, and the height of cylinder B is half that of cylinder A. Find the ratio of the curved surface area of A to the curved surface area of B. Also find the ratio of the volume of A to the volume of B.
Step-by-step solution
To find: CSA(A) : CSA(B) and V(A) : V(B)
Idea: Write both cylinders in terms of A’s radius r and height h, put them in the formulas and cancel. CSA has r once, but volume has r twice (r2), so doubling the radius affects the volume more.
- Let A have radius r and height h. Then B has radius 2r and height h2.½ mark
- CSA of A = 2πrh. CSA of B = 2π(2r)(h2) = 2πrh.½ mark
- Ratio of CSAs = 2πrh : 2πrh = 1 : 1.½ mark
- Volume of A = πr2h. Volume of B = π(2r)2(h2) = π × 4r2 × h2 = 2πr2h.1 mark
- Ratio of volumes = πr2h : 2πr2h = 1 : 2.½ mark
Check: Take r = 1, h = 4: A has CSA 8π, volume 4π; B (r = 2, h = 2) has CSA 8π, volume 8π. Ratios 1 : 1 and 1 : 2 ✓.
Answer to write in the exam
A: radius r, height h; B: radius 2r, height h2
CSA of ACSA of B = 2πrh2π(2r)(h/2) = 2πrh2πrh = 1
V of AV of B = πr2hπ(2r)2(h/2) = πr2h2πr2h = 12
∴ CSA ratio = 1 : 1; volume ratio = 1 : 2
Common mistakes that cost marks
- Writing (2r)2 = 2r2 instead of 4r2, which gives a volume ratio of 1 : 1.
- Writing the ratio the wrong way round (B : A).
How this can come in the exam
The radius of a cylinder is halved and its height is doubled. Its volume becomes
- the same
- half
- double
- one-fourth
Show answer
(B) half
π(r2)2(2h) = 12πr2h, half.
Try one yourself
Cylinder Q has three times the radius of cylinder P and one-third of its height. Find CSA(P) : CSA(Q) and V(P) : V(Q).
Show answer
CSA: 2πrh : 2π(3r)(h3) = 1 : 1. Volume: πr2h : π(9r2)(h3) = 1 : 3.
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