Question

From a point on a bridge across a river, the
angles of depression of the banks on opposite
sides at the river are 30° and 45°, respectively
If the bridge is at a height at 3 m from the
banks, find the width at the river.​

Answer 1

Step-by-step explanation:

width of river is 3(1+√3) m if we want we can use √3 value and write then the anwer will be 8.19 m

Answer 2

$\huge \boxed{ \fcolorbox{cyan}{pink}{Answer : }}$

$\sf{A \: and \: B \: represent \: points \: on \: the \: banks}$

$\sf{on \: opposite \: sides}$

so

$\rm{AB \: is \: width}$

Now

$\sf{Height \: of \: brigde \: DP = 3m}$

$\sf{AB = AD + DB}$

$\sf{In \: right \: triangle \: ABD, \: Angle A = 30°}$

Then

$\sf{ \tan(30) = \frac{pd}{ad}}$

$\rm{ \frac{1}{ \sqrt{3} } = \frac{3}{ad}}$

$\bf{ \huge{ \boxed{ \red{ \tt{AD= 3 \sqrt{3} \: }}}}}$

Then

$\sf{In \: triangle \: PBD, Angle \: B = 45°}$

$\sf{So \: BD = PD = 3m}$

now

$\rm{AB = DB + AD}$

$\sf{3 + 3 \sqrt{3}}$

$\sf{3( \sqrt{3 + 1)m}}$

$\rm{The \: width \: of \: river \: is}$

$\bf{ \huge{ \boxed{ \green{ \tt{3( \sqrt{3 + 1)m \: }}}}}}$