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Question 9 Find the length of the curve. x = 2 sin3 t, y = 2 cos3 t, ostsi O A) 4 OB) 12 OC) 2 OD 6

User Richardo
by
7.6k points

1 Answer

4 votes

The length of the curve given by the parametric equations x = 2sin(3t) and y = 2cos(3t) is 12 units.

Explanation: To find the length of the curve, we can use the arc length formula for parametric curves:

=

(

)

2

+

(

)

2

s=∫

a

b

(

dt

dx

)

2

+(

dt

dy

)

2

dt

where

s is the length of the curve and

t ranges from

a to

b.

In this case,

=

2

sin

(

3

)

x=2sin(3t) and

=

2

cos

(

3

)

y=2cos(3t), so we need to find

dt

dx

and

dt

dy

:

=

2

3

cos

(

3

)

=

6

cos

(

3

)

dt

dx

=2⋅3cos(3t)=6cos(3t)

=

2

3

sin

(

3

)

=

6

sin

(

3

)

dt

dy

=−2⋅3sin(3t)=−6sin(3t)

Now, we can plug these values back into the arc length formula and integrate:

=

(

6

cos

(

3

)

)

2

+

(

6

sin

(

3

)

)

2

s=∫

a

b

(6cos(3t))

2

+(−6sin(3t))

2

dt

=

36

cos

2

(

3

)

+

36

sin

2

(

3

)

s=∫

a

b

36cos

2

(3t)+36sin

2

(3t)

dt

=

36

(

cos

2

(

3

)

+

sin

2

(

3

)

)

s=∫

a

b

36(cos

2

(3t)+sin

2

(3t))

dt

=

36

s=∫

a

b

36

dt

=

6

s=6t

a

b

Conclusion: The length of the curve is

6

6

6b−6a. As we don't have the range of

t specified (from

a to

b), we cannot determine the exact length. However, if the range is from

=

0

t=0 to

=

2

t=2π (one full revolution), then

=

2

b=2π and

=

0

a=0, and the length would be

6

(

2

)

6

(

0

)

=

12

6(2π)−6(0)=12.

User Alejandro Aranda
by
8.5k points