Originally Posted by

**afeasfaerw23231233** p80 q11 ex18c

question:

A and D are two towns on the same side of the main road BC. A road APD is to be made to join A and D to the main road. where should the location of P be in order that the length of the road is its smallest? you can only use differentiation method.

my working:

let $\displaystyle l $ be the length of the road, x be the distance of BP

$\displaystyle l = \sqrt {x^2+4} +\sqrt {(10-x)^2+9} $

$\displaystyle dl/dx = \frac 1 2 (x^2+4)^{-\frac 1 2 } (2x) + \frac 1 2 (x^2 -20 x +109)^{-\frac 1 2 } (2x-20)$

$\displaystyle = x(x^2+4)^{-\frac 1 2 }+ (x-10)(x^2 - 20 x +109)^{-\frac 1 2 }$

i cannot find the value of x when dl/dx = 0 . thanks!