Question

Need help! (Calculus - Optimisation)

Two towns, Ancaster and Dundas, are 4 km and 6 km, respectively, from an old railroad line that has been made into a bike trail. Points C and D on the trail are the closest points to the two towns, respectively. These points are 8 km apart. Where should a rest stop be built to minimise the length of the new trail that must be built from both towns to the rest stop?

(Medals shall be awarded)

Answer 1

that is how i read the problem
the square of the distance between A and x is
\[x^2+16\] and the square of the distance between D and x is
\[(8-x)^2+36\]
minimize
\[x^2+16+(58-x)^2+36\]

Answer 2

i mean minimize
\[x^2+16+(8-x)^2+36\]

Answer 3

I see, thank you very much satellite :)

Answer 4

yw

Answer 5

btw you do not need calculus for this

Answer 6

\[x^2+16+(8-x)^2+36=2 x^2-16 x+116\] a quadratic
minimum is at the vertex, which you can find via \(-\frac{b}{2a}=-\frac{-16}{4}=4\)