A bicyclist is interested in the motion of a point on the tread of her tire, so she marks the tread at one point with chalk. She then rides the bike along a flat, level road. She imposes a coordinate system as shown in the figure below such that the chalk mark is touching the ground at the point (0,0) at timet=0. The radius of her wheels is 35 cm and the bike moves at 70cm/sec in the direction indicated. As the bike moves from left to right in the picture below, the chalk mark will trace out a path in the imposed coordinate system:

The parametric equations for the path of the chalk mark are

x(t) = 70t+ 35cos(2t/2)

y(t) = 35 + 35sin(2t/2)

A portion of the path (on the time interval [0,2]) is shown in this figure:

Give EXACT ANSWERS to all questions below; use pi to denote the number.

(a) The horizontal velocity is.

(b) The vertical velocity is.

(c) Computex'(t)at these times:

t = 0sec

t = 0.25sec

t = 0.5sec

t = 0.75sec

t = 1sec

(d) Computey'(t)at these times:

t = 0sec

t = 0.25sec

t = 0.5sec

t = 0.75sec

t = 1sec

(e) When is the first time the horizontal velocity will be a maximum?

(f) When is the first time the horizontal velocity will be a minimum?

(g) What is the maximum horizontal velocity?

(h) What is the minimum horizontal velocity?

(i) When is the first time the vertical velocity will be a maximum?

(j) When is the first time the vertical velocity be a minimum?

(k) What is the maximum vertical velocity?

(l) What is the minimum vertical velocity?

(m) The speed of the moving chalk mark at timetis defined by the equation

s(t) ={(x'(t))²+ (y'(t))²}.

Compute s(t).

(n) During the first 2 seconds, how many times will the speed be equal to 50 cm/sec?