vanka did not get a reply from grandfather even after two weeks. He becomes very sad. unable to bear the torture, He decides to write a letter to the coordinator, child Rights commission describing his terrible experiences. prepare the letter.
Answers
Answer:
\large{\mathbb{\colorbox{purple} {\boxed{\boxed{\colorbox{white} {-:Answer:-}}}}}}
-:Answer:-
\large{ \pmb{ \underline{ \underline{\frak{ \color{peru}{Given::}}}}}}
Given::
Given::
\pink{➠}{ \sf{ \bf{ \frac {dv}{dt}} = \sf1 {cm}^{3}/s \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:\:\: \: \: \: \: \: \: ...(i) }}➠
dt
dv
=1cm
3
/s...(i)
\large{ \pmb{ \underline{ \underline{\frak{ \color{pink}{To \: find::}}}}}}
Tofind::
Tofind::
\pink{➠}{ \sf{ Rate \: of \: decrease \: of \: slant \: height.}}➠Rateofdecreaseofslantheight.
\large{ \pmb{ \underline{ \underline{\frak{ \color{pin}{Formula \: used::}}}}}}
Formulaused::
Formulaused::
\pink{➠}{ \sf{ Volume_{(cone)}= \frac{1}{3}\pi {r}^{2}h }}➠Volume
(cone)
=
3
1
πr
2
h
\large{ \pmb{ \underline{ \underline{\frak{ \color{plum}{Construction::}}}}}}
Construction::
Construction::
\pink{➠}{ \sf{ Kindly \: refer \: the \: attachment\:also!!}}➠Kindlyrefertheattachmentalso!!
\large{ \pmb{ \underline{ \underline{\frak{ \color{blue}{Concept \: required::}}}}}}
Conceptrequired::
Conceptrequired::
\pmb{ \bf{From \: the \: above \: attachment..}}
Fromtheaboveattachment..
Fromtheaboveattachment..
\pink{➠}{ \sf{cos \: {\bf x} =\frac{h}{l} = \frac{ \sqrt{3} }{2} }}➠cosx=
l
h
=
2
3
\pink{➠}{ \sf{Radius_{(cone)},(r)= \frac{1}{2} }}➠Radius
(cone)
,(r)=
2
1
\pink{➠}{ \sf{Height_{(cone)},(h)= \frac{ \sqrt{3} }{2} l}}➠Height
(cone)
,(h)=
2
3
l
\large{ \pmb{ \underline{ \underline{\frak{ \color{cyan}{According \: to \: Question::}}}}}}
AccordingtoQuestion::
AccordingtoQuestion::
\pmb{ \bf{Let's \: start \: with \: help \: of \: formulas!!}}
Let
′
sstartwithhelpofformulas!!
Let
′
sstartwithhelpofformulas!!
\pink{➠}{ \sf{ Volume_{(cone)}= \frac{1}{3}\pi {r}^{2}h }}➠Volume
(cone)
=
3
1
πr
2
h
{: : \implies{ \sf{ Volume_{(cone)}= \frac{1}{3}\pi { \big( \frac{1}{2} \big)}^{2} \Big( \frac{ \sqrt{3} }{2}l \Big) }}}::⟹Volume
(cone)
=
3
1
π(
2
1
)
2
(
2
3
l)
{: : \implies{ \sf{ Volume_{(cone)}= \frac{\pi}{8 \sqrt{3} } }}}::⟹Volume
(cone)
=
8
3
π
\pink{➠}{ \sf{ \bf{ \frac {dv}{dt}} = \sf1 {cm}^{3}/s \: \: \: \: \: \: \:\{ from \: eq. \: ..(i) \}}}➠
dt
dv
=1cm
3
/s{fromeq...(i)}
: : \implies{ \sf{ \bf{ \frac {d}{dt}}} \Big[\frac{\pi}{8 \sqrt{3}} {l}^{3} \Big ] = \sf1 {cm}^{3}/s }::⟹
dt
d
[
8
3
π
l
3
]=1cm
3
/s
: : \implies{ \sf \frac{\pi}{8 \sqrt{3}} { \bf{ \frac {d}{dt}}}{(l)}^{3} = \sf1 {cm}^{3}/s }::⟹
8
3
π
dt
d
(l)
3
=1cm
3
/s
: : \implies{ \sf \frac{3 {l}^{2}\pi }{8 \sqrt{3}} { \bf{ \frac {d}{dt}}} = \sf1 {cm}^{3}/s }::⟹
8
3
3l
2
π
dt
d
=1cm
3
/s
{: : \implies{ \sf \frac{ \sqrt{3} (16\pi )}{8 } { \bf{ \frac {dl}{dt}}} = \sf1 {cm}^{3}/s \: \: \: \: \: \: \: \: \{∵At \: l=4cm \}}}::⟹
8
3
(16π)
dt
dl
=1cm
3
/s{∵Atl=4cm}
{: : \implies{ \sf2 \sqrt{3} \pi { \bf{ \frac {dl}{dt}}} = \sf1 {cm}^{3}/s}}::⟹2
3
π
dt
dl
=1cm
3
/s
{: : \implies{ \sf{ \bf{ \frac {dl}{dt}}} = \sf \: \frac{1}{2 \sqrt{3}\pi } {cm}^{3}/s}}::⟹
dt
dl
=
2
3
π
1
cm
3
/s
\pmb {\bf{Hence,}}
Hence,
Hence,
\purple᪣ {\boxed{ \sf{ Rate \: of \: decrease \: of \: slant \: height = \frac{1}{2 \sqrt{3} {\pi}}{cm}^{3} /s }}} ᪣᪣
Rateofdecreaseofslantheight=
2
3
π
1
cm
3
/s
᪣
ᚚᚚᚚᚚᚚᚚᚚᚚᚚᚚᚚᚚᚚᚚᚚᚚᚚᚚᚚᚚᚚᚚᚚᚚᚚᚚᚚᚚᚚᚚᚚᚚᚚᚚᚚᚚ
vanka did not get a reply from grandfather even after two weeks. He becomes very sad. unable to bear the torture, He decides to write a letter to the coordinator, child Rights commission describing his terrible experiences. prepare the letter.