Hi evry1, this is a problem i encountered and i cant seem to solve it. can you please help me out. I would really appreciate it if you could help me with all the questions, as im really struggling!!! thanks (sorry about the vertical lines, they are meant to be solid, but i cudnt make solid lines on the computer, and ignore the full stops, as they are included becos you cant just use spacing)

..........start boundary

S _____________________

|.angle Y........sand.........|

|..................................| 200meters

|------g------- X -------|

|..mud..........................| 200meters

|______________angle Z_|

........^ 500meters..........F

........^

........^

........^

........^

.....finish boundary

There is a race that goes through sand and mud, from S to F, passing through X as shown. Kylie can run at 4.5meters/second through sand and 2.5meters/second through mud. The track is the hypotenues from S to X, then the hypotenus from X to F. There is an angle Y, which is between the track (hypotenus from S to X) and the start boundary and also angle Z, which is between the track (hypotenus from X to F) and the finish boundary.

Question 1)

show that the total distance covered in the race, d=200(secY+secZ) and prove that Y and Z are also related by the equation tanY+tanZ=K, where 'K' is a constant.

Question 2)

Find an expression for d in terms of Y and state the implied domain of the corresponding function. Find the total distance kylie runs if she starts off at an angle of 50degrees(Y) and state the angleZ she must use when she reaches the second surface

Question 3)

Find an expression for t, the time taken for her to travel from S to F, in terms of Y and state the implied domain of the corresponding function. Plot the graph of t against g (the distance from the S vertical boundary and X) and comment on what the graph shows.

Question 4)

Another contestant, Emily, can run 4meters/second through sand and 3meters/second through mud. Determine Emily's best route, ensuring that she starts at point S and finishes at point F, passing through a point X, but it doesnt necessarily have to be the same point that Kylie passed through. Who wins the contest and by how many seconds?

Question 5)

Now let t1= the time taken to travel from S to X along the hypotenuse

and let t2= the time taken to travel from X to F along the hypotenuse

and t= t1 + t2

Express t1 and t2 in terms of 'g' and find dt1/dg (derivative) and dt2/dg (derivative) in terms of g

Question 6)

With the angles of Y and Z as shown originally in the figure, express dt1/dg and dt2/dg in terms of Y and Z respectively

Question 7)

Hence express dt/dg in terms of Y and Z and show that the minimum time occurs when (sinY/a) = (sinZ/b) where a and b are constants. Give an interpretation of these constants in terms of the situation being modelled.

Question 8)

Solve (sinY/a) = (sinZ/b) and tanY+tanZ=K to find the values of Y and Z which simultaneously satisfy each of these equations.

Question 9)

Hence verify that the best route for Kylie is the same as that found previously