Answer:
a. Ratio of their circumference = 2:3
b. The ratio of their surface areas = 2:3
c. The ratio of their volume = 4:9
Step-by-step explanation:
Let the cylinders be labelled as A and B respectively.
Diameter of cylinder A = 6 in,
its radius = [tex]\frac{diameter}{2}[/tex]
= [tex]\frac{6}{2}[/tex] = 3 in
Diameter of cylinder B = 9 in,
its radius = [tex]\frac{diameter}{2}[/tex]
= [tex]\frac{9}{2}[/tex] = 4.5 in
Thus,
a. circumference of a circle = 2[tex]\pi[/tex]r
Circumference of cylinder A = 2[tex]\pi[/tex]r
= 2 x [tex]\pi[/tex] x 3
= 6[tex]\pi[/tex] in
Circumference of cylinder B = 2[tex]\pi[/tex]r
= 2 x [tex]\pi[/tex] x 4.5
= 9[tex]\pi[/tex]
Ratio of their circumference = [tex]\frac{circumference of cylinder A}{circumference of cylinder B}[/tex]
= [tex]\frac{6\pi }{9\pi }[/tex]
= 2:3
b. The surface area of a cylinder = 2[tex]\pi[/tex]rh
the surface area of cylinder A = 2[tex]\pi[/tex]rh
= 6[tex]\pi[/tex]h
the surface area of cylinder B = 2[tex]\pi[/tex]rh
= 9[tex]\pi[/tex]h
The ratio of their surface areas = [tex]\frac{surface area of cylinder A}{surface area of cylinder B}[/tex]
= [tex]\frac{6\pi h}{9\pi h}[/tex]
= 2:3
c. Volume of a cylinder = [tex]\pi[/tex][tex]r^{2}[/tex]h
volume of cylinder A = [tex]\pi[/tex][tex]r^{2}[/tex]h
= [tex]\pi[/tex] x [tex]3^{2}[/tex] x h
= 9[tex]\pi[/tex]h
volume of cylinder b = [tex]\pi[/tex][tex]r^{2}[/tex]h
= [tex]\pi[/tex] x [tex]4.5^{2}[/tex] x h
= 20.25[tex]\pi[/tex]h
The ratio of their volume = [tex]\frac{volume of cylinder A}{volume of cylinder B}[/tex]
= [tex]\frac{9\pi h}{20.25\pi h}[/tex]
= 4:9
d=6in,9in
r=3in,4.5in
circumference:-
[tex]\\ \sf\longmapsto \dfrac{2\pi r_1}{2\pi r_2}[/tex]
[tex]\\ \sf\longmapsto \dfrac{6\pi}{9\pi}[/tex]
[tex]\\ \sf\longmapsto \dfrac{2}{3}[/tex]
[tex]\\ \sf\longmapsto 2:3[/tex]
LSA:-
[tex]\\ \sf\longmapsto \dfrac{2\pi r_1h}{2\pi r_2h}[/tex]
[tex]\\ \sf\longmapsto \dfrac{6}{9}[/tex]
[tex]\\ \sf\longmapsto \dfrac{2}{3}[/tex]
[tex]\\ \sf\longmapsto 2:3[/tex]
TSA
[tex]\\ \sf\longmapsto \dfrac{2\pi r_1(h+r_1)}{2\pi r_2(h+r_2)}[/tex]
[tex]\\ \sf\longmapsto \dfrac{6(h+6)}{9(h+9)}[/tex]
[tex]\\ \sf\longmapsto \dfrac{6h+36}{9h+81}[/tex]
[tex]\\ \sf\longmapsto \dfrac{h(6+36)}{h(9+81)}[/tex]
[tex]\\ \sf\longmapsto \dfrac{42}{90}[/tex]
[tex]\\ \sf\longmapsto \dfrac{7}{15}[/tex]
[tex]\\ \sf\longmapsto 7:15[/tex]
Volume:-
[tex]\\ \sf\longmapsto \dfrac{\pi r_1^2h}{\pi r_2^2h}[/tex]
[tex]\\ \sf\longmapsto \dfrac{3^2}{4.5^2}[/tex]
[tex]\\ \sf\longmapsto \dfrac{9}{20.25}[/tex]
[tex]\\ \sf\longmapsto 4:9[/tex]
