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Question Number 47449 by ajfour last updated on 10/Nov/18

Commented by ajfour last updated on 10/Nov/18

Find the area of the shadow of  the right circular cone (not  including its base area).  S is a point source of light, at  a distance d from center of  base of cone, at a height H.

$${Find}\:{the}\:{area}\:{of}\:{the}\:{shadow}\:{of} \\ $$$${the}\:{right}\:{circular}\:{cone}\:\left({not}\right. \\ $$$$\left.{including}\:{its}\:{base}\:{area}\right). \\ $$$${S}\:{is}\:{a}\:{point}\:{source}\:{of}\:{light},\:{at} \\ $$$${a}\:{distance}\:{d}\:{from}\:{center}\:{of} \\ $$$${base}\:{of}\:{cone},\:{at}\:{a}\:{height}\:{H}. \\ $$

Answered by mr W last updated on 10/Nov/18

Commented by mr W last updated on 10/Nov/18

((e+d)/e)=(H/h)  ⇒e=((hd)/(H−h))  cos θ=(r/e)=((r(H−h))/(hd))  ⇒θ=cos^(−1) ((r(H−h))/(hd))  sin θ=((√(h^2 d^2 −r^2 (H−h)^2 ))/(hd))  A_(shadow) =re sin θ−θπ=((rhd)/(H−h))×((√(h^2 d^2 −r^2 (H−h)^2 ))/(hd))−θr^2   ⇒A_(shadow) =((r(√(h^2 d^2 −r^2 (H−h)^2 )))/(H−d))−r^2 cos^(−1) ((r(H−h))/(hd))

$$\frac{{e}+{d}}{{e}}=\frac{{H}}{{h}} \\ $$$$\Rightarrow{e}=\frac{{hd}}{{H}−{h}} \\ $$$$\mathrm{cos}\:\theta=\frac{{r}}{{e}}=\frac{{r}\left({H}−{h}\right)}{{hd}} \\ $$$$\Rightarrow\theta=\mathrm{cos}^{−\mathrm{1}} \frac{{r}\left({H}−{h}\right)}{{hd}} \\ $$$$\mathrm{sin}\:\theta=\frac{\sqrt{{h}^{\mathrm{2}} {d}^{\mathrm{2}} −{r}^{\mathrm{2}} \left({H}−{h}\right)^{\mathrm{2}} }}{{hd}} \\ $$$${A}_{{shadow}} ={re}\:\mathrm{sin}\:\theta−\theta\pi=\frac{{rhd}}{{H}−{h}}×\frac{\sqrt{{h}^{\mathrm{2}} {d}^{\mathrm{2}} −{r}^{\mathrm{2}} \left({H}−{h}\right)^{\mathrm{2}} }}{{hd}}−\theta{r}^{\mathrm{2}} \\ $$$$\Rightarrow{A}_{{shadow}} =\frac{{r}\sqrt{{h}^{\mathrm{2}} {d}^{\mathrm{2}} −{r}^{\mathrm{2}} \left({H}−{h}\right)^{\mathrm{2}} }}{{H}−{d}}−{r}^{\mathrm{2}} \mathrm{cos}^{−\mathrm{1}} \frac{{r}\left({H}−{h}\right)}{{hd}} \\ $$

Commented by ajfour last updated on 10/Nov/18

Thank you Sir.

$${Thank}\:{you}\:{Sir}.\: \\ $$

Commented by mr W last updated on 10/Nov/18

thank you for checking sir!  can you figure it out if we have a  cylinder instead of a cone?

$${thank}\:{you}\:{for}\:{checking}\:{sir}! \\ $$$${can}\:{you}\:{figure}\:{it}\:{out}\:{if}\:{we}\:{have}\:{a} \\ $$$${cylinder}\:{instead}\:{of}\:{a}\:{cone}? \\ $$

Commented by ajfour last updated on 10/Nov/18

Commented by ajfour last updated on 10/Nov/18

cant figure properly, not sure,  Sir.

$${cant}\:{figure}\:{properly},\:{not}\:{sure}, \\ $$$${Sir}. \\ $$

Commented by MrW3 last updated on 11/Nov/18

image is nice and correct.  please solve it sir.

$${image}\:{is}\:{nice}\:{and}\:{correct}. \\ $$$${please}\:{solve}\:{it}\:{sir}. \\ $$

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