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Question Number 23564 by NECx last updated on 01/Nov/17

The curved surface area of a cone  is 21cm^2  .Calculate the curved  surface area of a similar cone  whose height is 4 times the other.

$${The}\:{curved}\:{surface}\:{area}\:{of}\:{a}\:{cone} \\ $$$${is}\:\mathrm{21}{cm}^{\mathrm{2}} \:.{Calculate}\:{the}\:{curved} \\ $$$${surface}\:{area}\:{of}\:{a}\:{similar}\:{cone} \\ $$$${whose}\:{height}\:{is}\:\mathrm{4}\:{times}\:{the}\:{other}. \\ $$

Commented by NECx last updated on 01/Nov/17

please show workings

$${please}\:{show}\:{workings} \\ $$

Commented by ajfour last updated on 01/Nov/17

radius also becomes 4 times or  radius stays the same ?

$${radius}\:{also}\:{becomes}\:\mathrm{4}\:{times}\:{or} \\ $$$${radius}\:{stays}\:{the}\:{same}\:? \\ $$

Commented by NECx last updated on 01/Nov/17

thats how the question is

$${thats}\:{how}\:{the}\:{question}\:{is} \\ $$

Commented by NECx last updated on 01/Nov/17

i dont know how possible it is but  if theres any possibility of the  answer , please help out.

$${i}\:{dont}\:{know}\:{how}\:{possible}\:{it}\:{is}\:{but} \\ $$$${if}\:{theres}\:{any}\:{possibility}\:{of}\:{the} \\ $$$${answer}\:,\:{please}\:{help}\:{out}. \\ $$

Answered by Joel577 last updated on 02/Nov/17

A  = πrs    (s = (√(t^2  + r^2 )))        = πr(√(t^2  + r^2 ))  (A_1 /A_2 ) = ((πr(√(t_1 ^2  + r^2 )))/(πr(√(t_2 ^2  + r^2 )))) = ((√(t_1 ^2  + r^2 ))/(√((4t_1 )^2  + r^2 )))  A_2  = (√((16t^2  + r^2 )/(t^2  + r^2 ))) . A_1

$${A}\:\:=\:\pi{rs}\:\:\:\:\left({s}\:=\:\sqrt{{t}^{\mathrm{2}} \:+\:{r}^{\mathrm{2}} }\right) \\ $$$$\:\:\:\:\:\:=\:\pi{r}\sqrt{{t}^{\mathrm{2}} \:+\:{r}^{\mathrm{2}} } \\ $$$$\frac{{A}_{\mathrm{1}} }{{A}_{\mathrm{2}} }\:=\:\frac{\pi{r}\sqrt{{t}_{\mathrm{1}} ^{\mathrm{2}} \:+\:{r}^{\mathrm{2}} }}{\pi{r}\sqrt{{t}_{\mathrm{2}} ^{\mathrm{2}} \:+\:{r}^{\mathrm{2}} }}\:=\:\frac{\sqrt{{t}_{\mathrm{1}} ^{\mathrm{2}} \:+\:{r}^{\mathrm{2}} }}{\sqrt{\left(\mathrm{4}{t}_{\mathrm{1}} \right)^{\mathrm{2}} \:+\:{r}^{\mathrm{2}} }} \\ $$$${A}_{\mathrm{2}} \:=\:\sqrt{\frac{\mathrm{16}{t}^{\mathrm{2}} \:+\:{r}^{\mathrm{2}} }{{t}^{\mathrm{2}} \:+\:{r}^{\mathrm{2}} }}\:.\:{A}_{\mathrm{1}} \\ $$

Commented by Rasheed.Sindhi last updated on 02/Nov/17

From above  (A_1 /A_2 ) = ((πr_1 (√(t_1 ^2  + r_1 ^2 )))/(πr_2 (√(t_2 ^2  + r_2 ^2 )))) = ((r_1 (√(t_1 ^2  + r_1 ^2 )))/((4r_1 )(√((4t_1 )^2  +( 4r_1 )^2 ))))                    = ((√(t_1 ^2  + r_1 ^2 ))/(4.4(√((t_1 )^2  +( r_1 )^2 ))))=(1/(16))  A_2 =16A_1 =16(21)=336 cm^2

$$\mathrm{From}\:\mathrm{above} \\ $$$$\frac{{A}_{\mathrm{1}} }{{A}_{\mathrm{2}} }\:=\:\frac{\pi{r}_{\mathrm{1}} \sqrt{{t}_{\mathrm{1}} ^{\mathrm{2}} \:+\:{r}_{\mathrm{1}} ^{\mathrm{2}} }}{\pi{r}_{\mathrm{2}} \sqrt{{t}_{\mathrm{2}} ^{\mathrm{2}} \:+\:{r}_{\mathrm{2}} ^{\mathrm{2}} }}\:=\:\frac{{r}_{\mathrm{1}} \sqrt{{t}_{\mathrm{1}} ^{\mathrm{2}} \:+\:{r}_{\mathrm{1}} ^{\mathrm{2}} }}{\left(\mathrm{4}{r}_{\mathrm{1}} \right)\sqrt{\left(\mathrm{4}{t}_{\mathrm{1}} \right)^{\mathrm{2}} \:+\left(\:\mathrm{4r}_{\mathrm{1}} \right)^{\mathrm{2}} }} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\frac{\sqrt{{t}_{\mathrm{1}} ^{\mathrm{2}} \:+\:{r}_{\mathrm{1}} ^{\mathrm{2}} }}{\mathrm{4}.\mathrm{4}\sqrt{\left({t}_{\mathrm{1}} \right)^{\mathrm{2}} \:+\left(\:\mathrm{r}_{\mathrm{1}} \right)^{\mathrm{2}} }}=\frac{\mathrm{1}}{\mathrm{16}} \\ $$$${A}_{\mathrm{2}} =\mathrm{16A}_{\mathrm{1}} =\mathrm{16}\left(\mathrm{21}\right)=\mathrm{336}\:\mathrm{cm}^{\mathrm{2}} \\ $$$$ \\ $$

Commented by Rasheed.Sindhi last updated on 02/Nov/17

If the height is 4 times,the radius  also should be 4 times because  the two cones are similar.

$$\mathrm{If}\:\mathrm{the}\:\mathrm{height}\:\mathrm{is}\:\mathrm{4}\:\mathrm{times},\mathrm{the}\:\mathrm{radius} \\ $$$$\mathrm{also}\:\mathrm{should}\:\mathrm{be}\:\mathrm{4}\:\mathrm{times}\:\mathrm{because} \\ $$$$\mathrm{the}\:\mathrm{two}\:\mathrm{cones}\:\mathrm{are}\:\boldsymbol{\mathrm{similar}}. \\ $$

Commented by NECx last updated on 02/Nov/17

wow!... thanks

$${wow}!...\:{thanks} \\ $$

Commented by NECx last updated on 02/Nov/17

could there be another approach  to solving this question

$${could}\:{there}\:{be}\:{another}\:{approach} \\ $$$${to}\:{solving}\:{this}\:{question} \\ $$$$ \\ $$

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