An object is moving in the plane according to these parametric equations:
x(t) = πt + cos(4πt + π/2)
y(t) = sin(4πt + π/2)
a portion of the "cycloidal" path (on the time interval [0,1]) is shown in this figure:

at time t=0, the object is located at (0,1). give exact answers to all questions below; use pi to denote the number π.
(a) the horizontal velocity is .
(b) the vertical velocity is .
(c) the slope of the tangent line to the path at time t is

(d) the equation of the tangent line to the path at time t=1/6 is
y = (x - ) +

(e) the first time the tangent line is vertical will be t=

(f) the second time the tangent line is vertical will be t=

(g) what is the maximum horizontal velocity?
(h) what is the minimum horizontal velocity?
(i) the speed of the object at time t is s(t)=

(j)compute s'(t)=

(k) the maximum speed is

(l) the minimum speed is

.a vx = -3π

b.vy = 0

c.c. m = sin(4πt + π/2) / [πt + cos(4πt + π/2)]

d.m = sin(4π/6 + π/2) / [π/6 + cos(4π/6 + π/2)]

e. t = -1.0

f.f. t = -0.35

g.vx = π - 4πsin (4π(0.124) + π/2)

h.vmax=4π cos (4π(0.045) + π/2)

i.s(t) = [x(t)^2 + y(t)^2]^(1/2)

s'(t) = d [x(t)^2 + y(t)^2]^(1/2) / dt

Explanation:

x(t) = πt + cos(4πt + π/2)

differentiating te orizontal distance wit respect to time t, will give horizontal velocity

change in displacement  per change in time is velocity

vx = dx/dt = π - 4πsin (4πt + π/2)

vx = π - 4π sin (0 + π/2)

at t =0, substituting te value of t into the above

vx = π - 4π (1)

vx = -3π

b,

y(t) = sin(4πt + π/2)

differentiate wit respect to t

dy/dt=4π cos (4πt + π/2)

π/2=90

when t=0

b. vy=dy/dt = 4π cos (4πt + π/2)

vy = 0

c.  slope of te tanent line

y(t)/x(t)

c. m = sin(4πt + π/2) / [πt + cos(4πt + π/2)]

d. at t=1/6, we substitute into answer gotten in c

m = sin(4π/6 + π/2) / [π/6 + cos(4π/6 + π/2)]

e. t = -1.0

f. t = -0.35

g. Solve for t  

vx = π - 4πsin (4πt + π/2) = 0

at maximum v=0

(4πt + π/2)=0.0043

t=-π/2+0.0043/(4π)

t=-0.124

vx = π - 4πsin (4π(0.124) + π/2)

h. Solve for t

vy = 4π cos (4πt + π/2) = 0

(4πt + π/2=1

t=0.57/4π

t=0.045

vmax=4π cos (4π(0.045) + π/2)

i.  resultant of te displacement

s(t) = [x(t)^2 + y(t)^2]^(1/2)

h. s'(t) = d [x(t)^2 + y(t)^2]^(1/2) / dt

s'(t)=

k and l. Solve for the values of t

d [x(t)^2 + y(t)^2]^(1/2) / dt = 0

And substitute to determine the maximum and minimum speeds.

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i had this question and the answer was 3.15 miles.

martina travels from school to her mom's office. from there, she walks to school. after school, she walks to the library and then to a friends house before finally ending back at home. find martina's total distance traveled.

a) 0 miles  

b) 0.4 miles  

c) 1 mile  

d) 3.15 miles