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Rom4ik [11]
3 years ago
9

Find the length of the arc of the circular helix with vector equation r(t) = 2 cos t i + 2 sin t j + tk from the point (2, 0, 0)

to the point (2, 0, 2π).
Mathematics
1 answer:
USPshnik [31]3 years ago
4 0

\vec r(t)=2\cos t\,\vec\imath+2\sin t\,\vec\jmath+t\,\vec k

\implies\vec r'(t)=-2\sin t\,\vec\imath+2\cos t\,\vec\jmath+\vec k

\implies\|\vec r'(t)\|=\sqrt{(-2\sin t)^2+(2\cos t)^2+1^2}=\sqrt5

Then the length of the arc is

\displaystyle\int_0^{2\pi}\|\vec r'(t)\|\,\mathrm dt=\sqrt5\int_0^{2\pi}\mathrm dt=2\sqrt5\,\pi

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4x + 3y = 6<br> 3x + 5y = -1<br> Solve the simultaneous equations.<br><br> y= <br> z=
Kisachek [45]

Answer:

x=3, y= -2

Step-by-step explanation:

Multiplying :

(4x+3y =6)*-3

(3x+5y =-1)*4

-12x -9y= -18

12x+20y= -4

Add the equations

20y-9y = -4-18

11y = -22

y= -22/11 = -2

3x +5* -2= -1

3x -10 = -1

3x = 10-1

x= 9/3 =3

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3 years ago
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Illusion [34]
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3 years ago
Unit rate for 3 inches of rain in 6 hours
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XYZ has coordinates X(2, 3), Y(1,4), and Z(8,9). A translation maps X to X'(4,7). What are the coordinates for Y' and Z' for thi
Rashid [163]

Answer:

The coordinates are Y'(x,y) = (3, 8) and Z'(x,y) = (10, 13).

Step-by-step explanation:

First, we have to derive an expression for translation under the assumption that each point of XYZ experiments the same translation. Vectorially speaking, translation from X to X' is defined by:

X'(x,y) = X(x,y) + T(x,y) (1)

Where T(x,y) is the vector translation.

If we know that X(x,y) = (2,3) and X'(x,y) = (4,7), then the vector translation is:

T(x,y) = X'(x,y)-X(x,y)

T(x,y) = (4,7) - (2,3)

T(x,y) = (2, 4)

Then, we determine the coordinates for Y' and Z':

Y'(x,y) = Y(x,y) + T(x,y)

Y'(x,y) = (1,4) + (2,4)

Y'(x,y) = (3, 8)

Z'(x,y) = Z(x,y) + T(x,y)

Z'(x,y) =(8,9) + (2,4)

Z'(x,y) = (10, 13)

The coordinates are Y'(x,y) = (3, 8) and Z'(x,y) = (10, 13).

7 0
3 years ago
A train traveled 317 miles in 10 hours. At this rate, how many miles will the train travel in 2 1/2 hours
Evgen [1.6K]

Answer: 79.25 miles

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2 years ago
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