There are a few optimisation and integration problems I can't seem to solve. Please help me, exam is on Tuesday... (Don't have to solve it, just tell me how to solve the questions )
1. P is a moveable point on the line y=7-x. Find the coordinates of P when it is closest to the point (2,4).
2. A small boat moving at x km/h uses fuel at a rate that is approximated by the function q=8+x^2/50, where q is measured in litres/hour. Determine the speech of the boat for which the amount of fuel used for any given journey is least.
3. An 8m high fence stands 3m from a large vertical wall. Find the length of the shortest ladder that will reach over the fence to the wall behind, as shown in the diagram.
4. A motorbike is 30km directly west of a car and begins moving east at 90km/h. At the same instant, the car begins moving north at 60km/h. What will be the minimum distance between the vehicles.
This is the integration question:
5. The cross-section of a channel is parabolic. It is 3 metres wide at the top and 2 metres deep. Find the depth of water, to the nearest cm, when the channel is half full. (For this one, I got that when the channel is half-full, 2 squared metres of the cross-section need to be covered, but I can seem to solve the question. Do I have to integrate with respect to y?)
Thank you so much for your help