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"Where should the point P be chosenon the line AB so as to maximize the angle theta"
how would i do this question?
will this work?: t = t1 + t2
tan t1=BP/BC (where C= the angle between B and P)
t1=arctan(BP/BC)
tan t2=AP/AD (where D= the angle between A and P)
t2=arctan(AP/AD)
sum= t1 + t2 --> arctan(BP/BC)+arctan(AP/AD)
then add in the known values:
t1 + t2= arctan(BP/2)+arctan(AP/5)
t= arctan((1/2)BP)+arctan((1/5)AP)
since we know AP+BP=3....solve for BP.
BP=-AP+3 and then substitute into the equation above:
t= arctan((1/2)(-AP+3))+arctan((1/5)AP)
t= -arctan((1/2AP-3/2))+arctan((1/5)AP)...then found the derivative
y= -1/(2(1+(1/2)AP-(3/2))^2) + 1/(5(1+(1/25)AP)^2)
i then solved for AP
and got 5+2sqrt(5)=9.472
and 5-2sqrt(5)=.5279
and my answer was .5279
is that correct?
*btw, AP+BP=3....because in the image it just shows the AP=3, when it should be that whole side=3.