Hi, I'm new to this forum. Any help would be greatly appreciated. I'm actually tutoring a refugee in a volunteer program, but it's been so long since I've done this maths that I'm having real trouble with it so in order to help her, I myself need some help!

A Queenland resort has a large swimming pool with AB=75. The pool is rectangular and has corners A,B,C,D, with A and C as opposites and B and D as opposites. P is a point somewhere between D and C.

A boy can swim at 1 m/s and run at 1, 2/3 m/s. He starts at A, swims to a point P on DC, and runs from P to C. He takes 2 seconds to pull himself out of the pool.

a, Let DP=x m and the total time be T s. Show that T=square root of (x squared +900)+3/5 (75-x)+2.

b, find dT/dx

c. i Find the value of x for which the time taken is a minimum.

ii Find the minimum time.

d. Find the time taken if the boy runs from A to D and then D to C.

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An isosceles trapezoid is inscribed in the parabolas y=4-x squared as illustrated (the parabola intersects the x axis at (-2,0) and (2,0) and is symmetrical.

a. show that the area of the trapezoid is :

1/2(4-x squared) (2x+4)

b, show that the trapezoid has its greatest area when x=2/3

c, Repeat with the parabolas y = a squared - x squared

i, show that the area, A, of the trapezoid = ( a squared - x squared) ( a + x)

ii, use the product rule to find dA/ dx.

iii, show that a maximum occurs when x = a/3.

If anymore information is needed, please ask!

Thanks so much.