The displacement (in centimeters) of a particle moving back and forth along a straight line is given by the equation of motion s = 5 sin πt + 2 cos πt, where t is measured in seconds. (round your answers to two decimal places.)
(a) find the average velocity during each time period.
(i) [1, 2]? cm/s
(ii) [1, 1.1]? cm/s
(iii) [1, 1.01]? cm/s
(iv) [1, 1.001]? cm/s
(b) estimate the instantaneous velocity of the particle when t = 1? cm/s

a.

i.   0  cm/s

ii.  49183 cm/s

iii. 4188.75 cm/s

iv. -673.8 cm/s

b. Instantaneous velocity is -900 cm/s

Step-by-step explanation:

Upon differentiating s = 5 sin πt + 2 cos πtEqn 1

derivative of sin πt is π cos πt

derivative of cos πt is -π sin πt

Therefore derivative of s = 5 sin πt + 2 cos πt is  

5π cos πt - 2π sin πtEqn 2

Substituting [1, 2] cm/s into Eqn 2

At t = 1 sec ; 5(180) cos 180 - 2(180) sin 180 = -900 cm/s

At t = 2 sec 5(180) cos 360 - 2(180) sin 360 = 900 cm/s

Average velocity = [900 + (-900)] ÷ 2 = 0

Substituting [1, 1.1] cm/s into Eqn 2

At t = 1 sec ; 5(180) cos 180 - 2(180) sin 180 = -900 cm/s

At t = 1.1 sec 5(180) cos 198 - 2(180) sin 198 = 99,266 cm/s

Average velocity = [99,266cm/s + (-900)] ÷ 2 = 49183 cm/s

Substituting [1, 1.01] cm/s into Eqn 2

At t = 1 sec ; 5(180) cos 180 - 2(180) sin 180 = -900 cm/s

At t = 1.1 sec 5(180) cos 181.8 - 2(180) sin 181.8 = 9277.5 cm/s

Average velocity = [9277.5 + (-900)] ÷ 2 = 4188.75 cm/s

Substituting [1, 1.001] cm/s into Eqn 2

At t = 1 sec ; 5(180) cos 180 - 2(180) sin 180 = -900 cm/s

At t = 1.1 sec 5(180) cos 180.18 - 2(180) sin 180.18 = -447.6 cm/s

Average velocity = [-447.6 + (-900)] ÷ 2 = -673.8 cm/s

b. Instantaneous velocity at t=1

   Substituting t=1 into Eqn 2

   At t = 1 sec ; 5(180) cos 180 - 2(180) sin 180 = -900 cm/s

55.44 inches

step-by-step explanation:

part a. assuming a perfectly linear relationship, we can find the slope from the first two data points.

slope = m = (change in rainfall)/(change in years)

= (61.80 -62.33)/(2005 -2004) = -0.53/1 = -0.53

then the point-slope form of the equation of the line can be written as

y = m(x -h) +k . . . . . m = -0.53, (h, k) = (0, 62.33)

y = -0.53x +62.33 . . . x = years after 2004

part b. in 2017, x = 2017 -2004 = 13. then the predicted rainfall is

y = -0.53·13 +62.33 = 55.44 . . inches

the predicted rainfall in 2017 is 55.44 in.

area of rectangle a is 2 times that of rectangle b that means 2x+x =   36cm^2

3x = 36cm ^2

x = 36/3

x= 12cm^2

and 2x = 24cm^2

now side of square = are divided by sides = 36/4 = 6cm

that means length of rectangle a and b is 6cm too

now for rectangle a,

l*b = area

= 6*y=24

y=24/6 = 4

therefore dimensions of rectangle a are- length = 6cm, breadth = 4cm

and for rectangle b, length = 6cm and breadth = 6-4 = 2cm.

hope this .