Question

100 POINTS!!!!!

Two cylinders are similar with diameters 6 in and 9 in.
a. Find the ratio of the circumference of their bases


b. Find the ratio of the surface areas.



c. Find the ratio of the volumes.

Answer 1

d=6in,9in

r=3in,4.5in

circumference:-

`\\ \sf\longmapsto (2\pi r_1)/(2\pi r_2)`

`\\ \sf\longmapsto (6\pi)/(9\pi)`

`\\ \sf\longmapsto (2)/(3)`

`\\ \sf\longmapsto 2:3`

LSA:-

`\\ \sf\longmapsto (2\pi r_1h)/(2\pi r_2h)`

`\\ \sf\longmapsto (6)/(9)`

`\\ \sf\longmapsto (2)/(3)`

`\\ \sf\longmapsto 2:3`

TSA

`\\ \sf\longmapsto (2\pi r_1(h+r_1))/(2\pi r_2(h+r_2))`

`\\ \sf\longmapsto (6(h+6))/(9(h+9))`

`\\ \sf\longmapsto (6h+36)/(9h+81)`

`\\ \sf\longmapsto (h(6+36))/(h(9+81))`

`\\ \sf\longmapsto (42)/(90)`

`\\ \sf\longmapsto (7)/(15)`

`\\ \sf\longmapsto 7:15`

Volume:-

`\\ \sf\longmapsto (\pi r_1^2h)/(\pi r_2^2h)`

`\\ \sf\longmapsto (3^2)/(4.5^2)`

`\\ \sf\longmapsto (9)/(20.25)`

`\\ \sf\longmapsto 4:9`

Answer 2

a. Ratio of their circumference = 2:3

b. The ratio of their surface areas = 2:3

c. The ratio of their volume = 4:9

Explanation:

Let the cylinders be labelled as A and B respectively.

Diameter of cylinder A = 6 in,

its radius =
`(diameter)/(2)`

=
`(6)/(2)` = 3 in

Diameter of cylinder B = 9 in,

its radius =
`(diameter)/(2)`

=
`(9)/(2)` = 4.5 in

Thus,

a. circumference of a circle = 2
`\pi`r

Circumference of cylinder A = 2
`\pi`r

= 2 x
`\pi` x 3

= 6
`\pi` in

Circumference of cylinder B = 2
`\pi`r

= 2 x
`\pi` x 4.5

= 9
`\pi`

Ratio of their circumference =
`(circumference of cylinder A)/(circumference of cylinder B)`

=
`(6\pi )/(9\pi )`

= 2:3

b. The surface area of a cylinder = 2
`\pi`rh

the surface area of cylinder A = 2
`\pi`rh

= 6
`\pi`h

the surface area of cylinder B = 2
`\pi`rh

= 9
`\pi`h

The ratio of their surface areas =
`(surface area of cylinder A)/(surface area of cylinder B)`

=
`(6\pi h)/(9\pi h)`

= 2:3

c. Volume of a cylinder =
`\pi`
`r^(2)`h

volume of cylinder A =
`\pi`
`r^(2)`h

=
`\pi` x
`3^(2)` x h

= 9
`\pi`h

volume of cylinder b =
`\pi`
`r^(2)`h

=
`\pi` x
`4.5^(2)` x h

= 20.25
`\pi`h

The ratio of their volume =
`(volume of cylinder A)/(volume of cylinder B)`

=
`(9\pi h)/(20.25\pi h)`

= 4:9