Two similar cylinders have lateral areas #81 \pi m^2 and 144 \pi m^2.# What is the ratios of (a) the heights, (b) the surface areas and (c) the volumes?

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Answer 1 · 2017-11-09T08:41:36

(a) ratio of heights#" "4:3#

(b) ratio of surface areas#" "16:9#

(c) ratio of volumes#" "64:27#

we need the relationships

#("ratio of areas")=("ratio of lengths")^2#

#("ratio of vols")=("ratio of lengths")^3#

let A#=#area;#" "V=#volume;#" "L=#length

we have

#A_2=144pim^2#

#A_1=81pim^3#

#:.A_2:A_1::144:81--(1)#

which cancels down to

#color(red)(A_2:A_1::16:9)#

from #(1)# we find ratio of lengths by square rooting

#:.L_2:L_1::sqrt(144:81)#

#:.L_2:L_1::12:9#

#color(red)(L_2:L_1::4:3)--(2)#

the ratio of vols is found by cubing #(2)#

#V_2:V_1::(4:3)^3#

#color(red)(V_2:V_1::64:27)#