Find the curved surface area of the cylinder where the ratio between the radius of the base and the height of cylindrical is 2:3 and its volume is1617cm3. A cylindrical pillar has volume 924m3 ad curved surface area 264m2. Find the height and diameter. Find the ratio of their radii of the cylinder to the pillar.

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1 : 100

100:1

200:1

1:200

answer is A.

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Detailed Solution

The curved surface area A of the cylinder is given by
A= πrh
The volume of the cylinder is given by
V= πr2h
Let assume the height of the cylinder as h1 and the radius of the cylinder as r1. We are given that the ratio between the radius of the base and the height of cylindrical is 2:3,

r 1 : h 1 = 2 : 3

\Rightarrow \frac{r 1}{h 1} = \frac{2}{3}

\Rightarrow h 1 = \frac{3}{2} r 1

We are also given in the question that the volume V of the cylinder is 1617cm3. We us the formula for volume of cylinder with r = r1, h = h1 and have,
V=1617

\Rightarrow π r_{1}^{2} h 1 = 1617

We put,

h 1 = \frac{3}{2} r 1

\Rightarrow \frac{22}{7} \times r 1 \times r 1 \times \frac{3}{2} \times r 1 = 1617

\Rightarrow r_{1}^{3} = 343

\Rightarrow r 1 = 7

So the radius of the cylinder is r1=7cm. We find the curved surface area, A of the cylinder in cm2 as,

A = 2 πr 1 h 1

= 2 \times π \times r 1 \times r 1 \times 32

= 2 \times 227 \times 7 \times 7 \times 32

= 462

We are also given that the cylindrical pillar has volume 924m3 and curved surface area 264m2. Let assume the height of the cylindrical pillar as h2 and the radius of the cylindrical pillar as r2.So we have from the formula of volume of cylinder,
V= πr22h2= 924
⇒ (πr2h2)r2 = 924...(1)
We have from the formula for curved surface area,
A= 2πr2h2 = 264
⇒πr2h2 = 2642 = 132..(2)
We put the value in equation(1) and get,
(πr2h2)r2= 924
⇒132r2= 924
⇒r2 = 924132= 7m
We know that the diameter is twice the length of radius which 2r2=2×7=14. We put r2 in equation (2) to get,
227×7×h2=132
⇒h2=6m
The ratio of the radii of the cylinder and the cylindrical pillar is
r1:r2=