CameraIcon
CameraIcon
SearchIcon
MyQuestionIcon


Question

Show that semi-vertical angle of right circular cone of given surface area and maximum volume is $$\sin ^{ -1 }{ \left( \cfrac { 1 }{ 3 } \right)  } $$.


Solution

$$h=r\text{cot}\theta$$
surface area is constant;
$$S=\pi r^{3}(1+\text{cosec}\theta)$$
$$\displaystyle \frac{dS}{dr}=0\implies\frac{d\theta}{dr}=\frac{2(1+\text{cosec}\theta)}{r(\text{cosec}\theta\text{cot}\theta)}$$
volume to be maximum ;
$$\displaystyle V=\frac{\pi r^{3}\text{cot}\theta}{3}\implies \frac{dV}{dr}=\frac{r^{2}\bigg(3\text{cot}\theta-r\text{cosec}^{2}\theta\displaystyle \frac{d\theta}{dr}\bigg)}{3}=0 \implies x=\text{sin}^{-1}\bigg(\frac{1}{3}\bigg)$$

985641_1078630_ans_83795b0da485458c9559d586c15b9f74.png

Maths

Suggest Corrections
thumbs-up
 
0


similar_icon
Similar questions
View More


similar_icon
People also searched for
View More



footer-image