Question

Semi vertical angle of a right circular cone formula

Answer 1

 Area =πrl+πr2πrl+πr2V=13V=13πr2hπr2h

Step 1:

Let rr be the radius ll be the slant height and hh be the vertical height of a cone of semi-vertical angle αα

Surface area S=πrl+πr2S=πrl+πr2------(1)

l=S−πr2πrl=S−πr2πr

The volume of the cone V=13V=13πr2hπr2h

=13=13πr2l2−r2−−−−−√πr2l2−r2

=πr23=πr23(S−πr2)2π2r2−r2−−−−−−−−−−−−√(S−πr2)2π2r2−r2

=πr23=πr23(S−πr2)2−π2r4π2r2−−−−−−−−−−−√(S−πr2)2−π2r4π2r2

=πr23=πr23S2−2πSr2+π2r4−π2r4−−−−−−−−−−−−−−−−−−√πrS2−2πSr2+π2r4−π2r4πr

=r3=r3S2−2πSr2+π2r4−π2r4−−−−−−−−−−−−−−−−−−−−−√S2−2πSr2+π2r4−π2r4

=r3=r3S(S−2πr2)−−−−−−−−−−√

Answer 2

Surface Area =πrl+πr2

V=13πr2h

***Step 1:

Let r be the radius l be the slant height and h be the vertical height of a cone of semi-vertical angle α

Surface area S=πrl+πr2------(1)

l=S−πr2πr

The volume of the cone V=13πr2h

=13πr2√l2−r2

=πr23√(S−πr2)2π2r2−r2

=πr23√(S−πr2)2−π2r4π2r2

=πr23√S2−2πSr2+π2r4−π2r4πr

=r3√S2−2πSr2+π2r4−π2r4

=r3√S(S−2πr2)

***Step 2:

V2=r29S(S−2πr2)

V2=S9(Sr2−2πr4)

dV2dr=S9[2Sr−8πr3]

d2V2dr2=S9[2S−24πr2]------(2)

Now dV2dr=0

⇒S9(2Sr−8πr3)=0

⇒(S−4πr2)=0

Putting S=4πr2 in (2)

d2Vdr2=S9[8πr2−24πr2]=−ve

⇒V is maximum when S=4πr2

***Step 3:

Putting the value in equ(1)

4πr2=πrl+πr2

4πr2−πr2=πrl

3πr2=πrl

3r2=rl

rl=13

sinα=13

l=sin−1(13)

Thus V is maximum when S=constant

α=sin−113

MAY THIS HELP U MY FRND!!!!