Optimization of a model concerning distance and time. Very stuck!!!?

Question

Town A is 30 kms east of town B. "Jo" drives to A twice a week, and B once a week. Where is the best place to have a house built so that travelling time will be minimised?
Assuming that house can be built anywhere on the straight line between A and B, I need to devise a formula, differentiate and set to 0 so that I can find the optimum place to build. The problem is, my formulas either differentiate to a number value or involve "speed" which is unknown. I'm missing something somewhere and would appreciate any suggestions!
Once I've solved this, I then have to take into account two journeys to town C situated 50kms north of town B, so the house would be built somwhere in the right angled triangle ABC.
Thanks in advance to anyone who could point me in the right direction!

Answer 1

♣ you should start with triangle ABC at once without solving useless problem of AB line. Have you caught it? If not, click me.

Answer 2

Actually, for the first part the answer will be at a boundary point, namely in Town A. Think about it.

But to make it formal, let x be the distance from A and S be the speed.

Then you're minimizing 2x/S + (30 - x)/S.

Since S is a constant here, carrying it around will not change your answer. And it's OK to point that out and say "Without loss of generality, S = 1"

For the more complex problem, you have two variables, the x and y coordinates, and a more complicated function -- twice the distance to A plus twice the distance to C plus the distance to B.

Just set that up, take the partial derivatives with respect to x and y, and hope that everything goes well.