Answer:
The volume of the smaller paint can is 720.85 [tex]cm^{3}[/tex].
Step-by-step explanation:
Since the two paint cans are similar, the dimensions of their diameters and heights are in a definite proportion.
The volume of a cylinder = [tex]\pi[/tex][tex]r^{2}[/tex]h
Since the diameter of the larger paint can is 64 cm, the radius = [tex]\frac{diameter}{2}[/tex]
= [tex]\frac{64}{2}[/tex]
= 32 cm
Given that the volume is 9600 cubic centimetre, the height of the larger paint can could be determined by;
V = [tex]\pi[/tex][tex]r^{2}[/tex]h
9600 = [tex]\frac{22}{7}[/tex] × [tex]32^{2}[/tex] × h
9600 = 3218.2857 h
⇒ h = 2.983 cm
Two cylinders are similar if their diameter and height are proportional.
Let [tex]d_{1}[/tex] represent the diameter of the larger paint can, [tex]d_{2}[/tex] the diameter of the smaller paint can, [tex]h_{1}[/tex] the height of the larger paint can and [tex]h_{2}[/tex] the height of the smaller paint can. So that;
[tex]\frac{d_{1} }{d_{2} }[/tex] = [tex]\frac{h_{1} }{h_{2} }[/tex]
[tex]\frac{64}{27}[/tex] = [tex]\frac{2.983}{h_{2} }[/tex]
⇒ [tex]h_{2}[/tex] = 1.2585
[tex]h_{2}[/tex] = 1.3 cm
If the diameter of the smaller paint can is 27, then its radius = 13.5 cm. So that its volume cane be determined by;
volume = [tex]\pi[/tex][tex]r^{2}[/tex]h
= [tex]\frac{22}{7}[/tex] × [tex](13.5)^{2}[/tex] ×1.2585
= 720.8508
= 720.85 [tex]cm^{3}[/tex]
The volume of the smaller paint can is 720.85 [tex]cm^{3}[/tex].
