Question and Answers Forum

All Questions      Topic List

Geometry Questions

Previous in All Question      Next in All Question      

Previous in Geometry      Next in Geometry      

Question Number 33452 by ajfour last updated on 16/Apr/18

Commented by ajfour last updated on 16/Apr/18

Find length of shadow s of a stick of length l when source of light is a point source at height H above ground.

$${Find}\:{length}\:{of}\:{shadow}\:{s}\:{of}\:{a}\: \\ $$$${stick}\:{of}\:{length}\:\boldsymbol{{l}}\:{when}\:{source} \\ $$$${of}\:{light}\:{is}\:{a}\:{point}\:{source}\:{at} \\ $$$${height}\:{H}\:{above}\:{ground}. \\ $$

Commented by ajfour last updated on 19/Apr/18

question for you, mrW Sir...

$${question}\:{for}\:{you},\:{mrW}\:{Sir}... \\ $$$$ \\ $$

Answered by MrW3 last updated on 12/Jun/18

Commented by MrW3 last updated on 12/Jun/18

x_1 =0, y_1 =H x_2 =u, y_2 =v x_3 =u+l cos α, y_3 =v+l sin α eqn. of line 12: ((x−0)/(y−H))=((u−0)/(v−H)) ⇒ x=(u/(v−H))(y−H) point 4: x_4 =(u/(v−H))(0−H)=((uH)/(H−v)) eqn. of line 13: ((x−0)/(y−H))=((u+l cos α−0)/(v+l sin α−H)) ⇒x=((u+l cos α)/(v+l sin α−H))(y−H) point 5: x_5 =((u+l cos α)/(v+l sin α−H))(0−H)=(((u+l cos α)H)/(H−v−l sin α)) s=x_5 −x_4 =(((u+l cos α)H)/(H−v−l sin α))−((uH)/(H−v)) =(((u+l cos α)(H−v)−u(H−v−l sin α))/((H−v−l sin α)(H−v)))×H ⇒s=(((H−v) cos α+u sin α)/((H−v−l sin α)(H−v)))×lH or ⇒s=(((H−y) cos α+x sin α)/((H−y−l sin α)(H−y)))×lH

$${x}_{\mathrm{1}} =\mathrm{0},\:{y}_{\mathrm{1}} ={H} \\ $$$${x}_{\mathrm{2}} ={u},\:{y}_{\mathrm{2}} ={v} \\ $$$${x}_{\mathrm{3}} ={u}+{l}\:\mathrm{cos}\:\alpha,\:{y}_{\mathrm{3}} ={v}+{l}\:\mathrm{sin}\:\alpha \\ $$$${eqn}.\:{of}\:{line}\:\mathrm{12}: \\ $$$$\frac{{x}−\mathrm{0}}{{y}−{H}}=\frac{{u}−\mathrm{0}}{{v}−{H}} \\ $$$$\Rightarrow\:{x}=\frac{{u}}{{v}−{H}}\left({y}−{H}\right) \\ $$$${point}\:\mathrm{4}: \\ $$$${x}_{\mathrm{4}} =\frac{{u}}{{v}−{H}}\left(\mathrm{0}−{H}\right)=\frac{{uH}}{{H}−{v}} \\ $$$${eqn}.\:{of}\:{line}\:\mathrm{13}: \\ $$$$\frac{{x}−\mathrm{0}}{{y}−{H}}=\frac{{u}+{l}\:\mathrm{cos}\:\alpha−\mathrm{0}}{{v}+{l}\:\mathrm{sin}\:\alpha−{H}} \\ $$$$\Rightarrow{x}=\frac{{u}+{l}\:\mathrm{cos}\:\alpha}{{v}+{l}\:\mathrm{sin}\:\alpha−{H}}\left({y}−{H}\right) \\ $$$${point}\:\mathrm{5}: \\ $$$${x}_{\mathrm{5}} =\frac{{u}+{l}\:\mathrm{cos}\:\alpha}{{v}+{l}\:\mathrm{sin}\:\alpha−{H}}\left(\mathrm{0}−{H}\right)=\frac{\left({u}+{l}\:\mathrm{cos}\:\alpha\right){H}}{{H}−{v}−{l}\:\mathrm{sin}\:\alpha} \\ $$$$ \\ $$$${s}={x}_{\mathrm{5}} −{x}_{\mathrm{4}} =\frac{\left({u}+{l}\:\mathrm{cos}\:\alpha\right){H}}{{H}−{v}−{l}\:\mathrm{sin}\:\alpha}−\frac{{uH}}{{H}−{v}} \\ $$$$=\frac{\left({u}+{l}\:\mathrm{cos}\:\alpha\right)\left({H}−{v}\right)−{u}\left({H}−{v}−{l}\:\mathrm{sin}\:\alpha\right)}{\left({H}−{v}−{l}\:\mathrm{sin}\:\alpha\right)\left({H}−{v}\right)}×{H} \\ $$$$\Rightarrow{s}=\frac{\left({H}−{v}\right)\:\mathrm{cos}\:\alpha+{u}\:\mathrm{sin}\:\alpha}{\left({H}−{v}−{l}\:\mathrm{sin}\:\alpha\right)\left({H}−{v}\right)}×{lH} \\ $$$${or} \\ $$$$\Rightarrow{s}=\frac{\left({H}−{y}\right)\:\mathrm{cos}\:\alpha+{x}\:\mathrm{sin}\:\alpha}{\left({H}−{y}−{l}\:\mathrm{sin}\:\alpha\right)\left({H}−{y}\right)}×{lH} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com