Hey guys, please help me with some of these questions, I'm really clueless how to solve them. Any help is very much appreciated.
1: A ship S leaves a Port P at 2 pm and it sails in the direction 30°T at 12km/h. Another ship is located 100 km East of P and is sailing towards P at 8km/h. (t is the number of hours after 2 pm)
a: Show that the distance D(t), between the two ships is given by D(t) = Square root of: 304t²-2800t+10000
b: Find the minimum value of [D(t)]² for all t>0.
c: At what time are both ships closest?
2:A fence AB is 1m high and 2m away from a wall, RQ. An ladder PQ is placed on the ence and touches the ground at P and the wall at Q.
a: If AP = x, find the value of QR in terms of x.
b: The ladder has the length L(x), show that [L(x)]² = (x+2)²(1+(1/x²)
Show that derivative[L(x)]² with respect to x = 0, only when x = third square root of 2.
d: Find the shortest length of the ladder and prove that it is the shortest length.
3: A (symmetrical) gutter is made from a sheet of metal (30 cm wide) by bending it twice. For alpha as indicated:
a: Show that the cross-sectional area is given by A = 100cosalpha(1+cosalpha)
b: Using the result from a, show that derivative A with respect to alpha = 0 when sin alpha = 0.5 or -1.
c: What value has alpha when the gutter has its maximum carrying capacity?
Here I made a sketch for 2 of the problems: