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Question-83524

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Question Number 83524 by liki last updated on 03/Mar/20
Commented by liki last updated on 03/Mar/20
...help me plz qns no. 5,7 and 9
$$…{help}\:{me}\:{plz}\:{qns}\:{no}.\:\mathrm{5},\mathrm{7}\:{and}\:\mathrm{9} \\ $$
Commented by liki last updated on 03/Mar/20
...plz help me .....
$$…{plz}\:{help}\:{me}\:\:….. \\ $$
Answered by Kunal12588 last updated on 03/Mar/20
Commented by Kunal12588 last updated on 03/Mar/20
(dr_2 /dt)=1 cm/s , (dr_1 /dt)=2 cm/s A_(shaded) =πr_2 ^2 −πr_1 ^2 ⇒A_(shaded) =π(r_2 ^2 −r_1 ^2 ) ⇒(dA/dt)=2π(r_2 ×(dr_1 /dt)−r_1 ×(dr_2 /dt)) ⇒(dA/dt)=2π(r_2 (1)−r_1 (2)) ⇒(dA/dt)=2π(r_2 −2r_1 ) when r_1 =8 cm, r_2 =12 cm (dA/dt)=2π(12−16) (dA/dt)=−8π cm/s
$$\frac{{dr}_{\mathrm{2}} }{{dt}}=\mathrm{1}\:{cm}/{s}\:\:,\:\:\frac{{dr}_{\mathrm{1}} }{{dt}}=\mathrm{2}\:{cm}/{s} \\ $$$${A}_{{shaded}} =\pi{r}_{\mathrm{2}} ^{\mathrm{2}} −\pi{r}_{\mathrm{1}} ^{\mathrm{2}} \\ $$$$\Rightarrow{A}_{{shaded}} =\pi\left({r}_{\mathrm{2}} ^{\mathrm{2}} −{r}_{\mathrm{1}} ^{\mathrm{2}} \right) \\ $$$$\Rightarrow\frac{{dA}}{{dt}}=\mathrm{2}\pi\left({r}_{\mathrm{2}} ×\frac{{dr}_{\mathrm{1}} }{{dt}}−{r}_{\mathrm{1}} ×\frac{{dr}_{\mathrm{2}} }{{dt}}\right) \\ $$$$\Rightarrow\frac{{dA}}{{dt}}=\mathrm{2}\pi\left({r}_{\mathrm{2}} \left(\mathrm{1}\right)−{r}_{\mathrm{1}} \left(\mathrm{2}\right)\right) \\ $$$$\Rightarrow\frac{{dA}}{{dt}}=\mathrm{2}\pi\left({r}_{\mathrm{2}} −\mathrm{2}{r}_{\mathrm{1}} \right) \\ $$$${when}\:{r}_{\mathrm{1}} =\mathrm{8}\:{cm},\:{r}_{\mathrm{2}} =\mathrm{12}\:{cm} \\ $$$$\frac{{dA}}{{dt}}=\mathrm{2}\pi\left(\mathrm{12}−\mathrm{16}\right) \\ $$$$\frac{{dA}}{{dt}}=−\mathrm{8}\pi\:{cm}/{s} \\ $$
Answered by Kunal12588 last updated on 03/Mar/20
Commented by Kunal12588 last updated on 03/Mar/20
(r/h)=((12)/(18)) ⇒r=((2h)/3) V=(1/3)πr^2 h ⇒V=(1/3)π×(4/9)h^3 ⇒V=(4/(27))πh^3 ⇒(dV/dh)=(4/9)πh^2 ×(dh/dt) ⇒(dV/dh)=(4/9)π×36×(dh/dt) ⇒2=16π(dh/dt) ⇒(dh/dt)=(1/(8π))cm/s
$$\frac{{r}}{{h}}=\frac{\mathrm{12}}{\mathrm{18}} \\ $$$$\Rightarrow{r}=\frac{\mathrm{2}{h}}{\mathrm{3}} \\ $$$${V}=\frac{\mathrm{1}}{\mathrm{3}}\pi{r}^{\mathrm{2}} {h} \\ $$$$\Rightarrow{V}=\frac{\mathrm{1}}{\mathrm{3}}\pi×\frac{\mathrm{4}}{\mathrm{9}}{h}^{\mathrm{3}} \\ $$$$\Rightarrow{V}=\frac{\mathrm{4}}{\mathrm{27}}\pi{h}^{\mathrm{3}} \\ $$$$\Rightarrow\frac{{dV}}{{dh}}=\frac{\mathrm{4}}{\mathrm{9}}\pi{h}^{\mathrm{2}} ×\frac{{dh}}{{dt}} \\ $$$$\Rightarrow\frac{{dV}}{{dh}}=\frac{\mathrm{4}}{\mathrm{9}}\pi×\mathrm{36}×\frac{{dh}}{{dt}} \\ $$$$\Rightarrow\mathrm{2}=\mathrm{16}\pi\frac{{dh}}{{dt}} \\ $$$$\Rightarrow\frac{{dh}}{{dt}}=\frac{\mathrm{1}}{\mathrm{8}\pi}{cm}/{s} \\ $$

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