assignment 10 Query

#$&*

course Mth 164

4/13 5:30

If your solution to stated problem does not match the given solution, you should self-critique per instructions at

http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm

.

Your solution, attempt at solution. If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it. This response should be given, based on the work you did in completing the assignment, before you look at the given solution.

SOLUTIONS/COMMENTARY ON QUERY 10

Query problem 8.1.28 polar coordinates of (-3, 4`pi)

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10:38:28

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Your solution:

We have the polar coordinates of (-3, 4pi) so other polar coordinates of the point for which:

r > 0, -2pi <= theta < 0 is (3, -pi)

r < 0, 0 <= theta < 2pi is (-3, 0)

r > 0, 2pi <= theta < 4pi is (3, 3pi).

confidence rating #$&*: 3

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Given Solution:

** (-3, 4 pi) corresponds to 2 complete revolutions, corresponding to the angle 4 pi, which directs you along the positive x axis.

• However r = -3 indicates that we move 3 units in the opposite direction, so we'll end up 3 units to the left of the origin and on the x axis.

This point could also be described by the polar coordinates (3, pi) or (3, -pi), or (-3, 0). **

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Self-critique (if necessary):

OK

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Self-critique Rating: OK

**** Query problem 8.1.42 rect coord of (-3.1, 182 deg)

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Your solution:

Given the polar coordinates of (-3.1, 182 deg), the rectangular coordinates can be figured as follows:

x = -3.1cos(182 deg) or approx. 3.1

y = -3.1sin(182 deg) or approx. .11.

The rectangular coordinates are approximately (3.1, .11).

confidence rating #$&*: 3

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Given Solution:

GOOD STUDENT SOLUTION:

since we know the formula for converting polar coordinates to rectangular

coordinates we can say that r=-3.1 and theta=182 deg. we can first find the

value of x by saying

x=r cos theta which gives

(-3.1) cos 182 deg. = 3.1.

We can then find y by saying y= r sin theta so y=(-3.1)sin 182 deg. using a

calculator we get an approx. value of .1081.

Thus the rectangular coordinates are (3.1, .1081), approx..

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Self-critique (if necessary):

OK

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Self-critique Rating: OK

Query problem 8.1.54 polar coordinates of (-.8, -2.1)

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18:50:13

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Your solution:

Given the rectangular coordinates of (-.8, -2.1), the polar coordinates can be figured as follows:

r = sqrt((-.8)^2 + (-2.1)^2) or approx. 2.25

tan(theta) = y/x = -2.1 / -.8 so

theta = tan^-1 (-2.1 / -.8)¬¬ or approx. 1.21 or approx. 69.1 deg¬¬

Since x is negative, we add pi to the angle as follows:

69.1 deg + 180 deg = 249 .1 deg

The polar coordinates are (2.25, 249.1 deg).

confidence rating #$&*: 3

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Given Solution:

STUDENT SOLUTION:

For the rectangular coordinates of (-.8, -2.1) to find the polar coordinates:

we know that r^2 = x^2 + y^2

so thus we know r = sqrt( (-.8) ^2 + (-2.1) ^2)

so r = sqrt( .64 + 4.41)

r = sqrt 5.05

r = 2.247.

To find theta we use tan (theta) = y / x

So tan(theta) = (-2.1)/ (-.8)

tan (theta) = 2.625 and x < 0 so

theta = 69.145 deg + 180 deg = 249.145 deg

or about 249 degrees.

Thus the polar coordinate would be as follows,

( 2.2, 249 deg). **

** In radians we have arctan(2.625) = 1.21; adding pi radians because x < 0 we get 4.35 rad.

So the coordinates are (2.2, 4.35), approx.. **

This point is in the third quadrant so the angle would be pi + 1.21 rad, or 69.1 deg + 180 deg.

When the x component is negative the angle is in the second or third quadrant; the range of the arctan is the fourth and first quadarnt so when x is negative you need to add pi rad or 180 deg to arctan(y/x).

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Self-critique (if necessary):

OK

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Self-critique Rating: OK

**** Query problem 8.1.60 write y^2 = 2 x using polar coordinates.

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19:05:48

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Your solution:

y^2 = 2x = r^2*sin^2(theta) = 2r cos(theta)

= r^2*sin^2(theta) - 2r cos(theta) = 0

= r(r sin^2(theta) - 2 cos(theta) = 0

confidence rating #$&*: 3

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Given Solution:

** We can rearrange the equation to give the form

y^2 - 2 x = 0. Then since y = r sin (theta) and x = r cos (theta) we have

(r^2 sin ^2 theta) - ( 2 r cos (theta) = 0.

We can factor out r to get

r [r sin^2(theta) - 2 cos(theta) ] = 0,

which is equivalent to

r = 0 or r sin^2(theta) - 2 cos(theta) = 0.

The latter form can be solved for r. We get

r = 2 cos(theta) / sin^2(theta).

This form is convenient for graphing. **

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Self-critique (if necessary):

OK

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Self-critique Rating: OK

**** Query problem 8.1.52 exact polar coordinates of (-2, -2`sqrt(3))

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10:58:03

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Your solution:

Given the rectangular coordinates of (-2, -2`sqrt(3)), the polar coordinates can be figured as follows:

r = sqrt((sqrt(3))^2 + (1)^1) = sqrt(4 + 12) = 4

tan(theta) = -2sqrt(3) / -2 = sqrt(3) so

theta = tan^-1(sqrt(3)) = pi/3

Since rectangular coordinates are in the third quadrant, we either add pi to theta or use the ray extending the 4 units in the direction opposite the terminal side of theta for the polar coordinates are therefore (-4, pi/3).

confidence rating #$&*: 3

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Given Solution:

r^2=x^2+y^2 so

r^2=(-2)^2+(-2 sqrt(3))^2 = 4+12 = 16 so

so r=4.

tan theta= y/x = (-2 sqrt(3))/(-2) = sqrt(3).

This occurs for the basic angle theta = pi/3, and also for theta = pi/3 + pi = 4 pi/3.

The given point is in the third quadrant, so the angle is 4 pi/3.

The polar coordinates are therefore (4, 4 pi/3).

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Self-critique (if necessary):

OK

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Self-critique Rating: OK

**** Query problem 8.1.62 write 4 x^2 y = 1 using polar coordinates.

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11:05:39

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Your solution:

4x^2 y = 1

4(r cos(theta))^2 * r sin(theta) = 1 (x = rcos(theta), y= rsin(theta)

4 * r * r * cos(theta) * cos(theta) * r *sin(theta) = 1

4r^3 cos(theta) * cos(theta) * sin(theta) = 1 (combine r terms)

4 r^3 * cos(theta) * sin(2theta) = 1 (double angle formula)

confidence rating #$&*: 3

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Given Solution:

GOOD STUDENT SOLUTION

we can first say that since x=r cos theta and y= r sin theta that 4(r cos

theta)^2(r sin theta)=1

we then have

4 r^3 cos theta sin theta=1

then using the double angle formula we have

2r^3(sin 2 theta)=1

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11:05:40

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Self-critique (if necessary):

OK

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Self-critique Rating: OK

**** Query problem 8.1.72 rect coord form of r = 3 / (3 -

cos(`theta))

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Your solution:

r = 3 / (3 - cos(`theta))

r(3- cos(`theta)) = 3

3r - r cos(theta) = 3

3r - x = 3

3r = 3 + x

r = 1 + x/3

r^2 = (1 + 1/3 x)^2

r^2 = 1/9 x^2 + 2/3 x + 1

x^2 + y^2 = 1/9 x^2 + 2/3 x + 1

x^2 - 1/9 x^2 - 2/3 x + y^2 = 1

8/9 x^2 -2/3x + y^2 = 1

8/9(x^2 - 3/4x) + y^2 = 1

8/9(x - 3/8)^2 + y^2 = 9/8

Very good, but your next-to-last step is very close to the standard form for an ellipse, which is

(x - h)^2 / a^2 + (y - k)^2 / b^2 = 1

In this case h = 3/8, k = 0, a = sqrt(9/8) = 3 sqrt(2) / 4 and b = 1.

This is an ellipse with semiaxes a and b, centered at (3/8, 0).

This is almost, but not quite right. When you completed the square, you neglected to add (3/8)^2 to both sides.

I've corrected the given solution in a note after your self-critique.

confidence rating #$&*: 2

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Given Solution:

Multiply both sides of the equation by 3-cos(theta) to get

3r - r cos(theta)=3.

Substitute sqrt(x^2+y^2) for r and x for r cos(theta) to get

3sqrt(x^2+y^2)-x=3. Add x to both sides to obtain

3sqrt(x^2+y^2)=3 + x and square both sides:

3(x^2 + y^2) = x^2 + 6 x + 9, which simplifies to

2 x^2 + y^2 - 6x - 9 = 0.

Completing the square on 2 x^2 - 6x we get

2( x^2 - 3 x + 9/4 - 9/4 ) + y^2 = 9 so

2 ( x+3/2)^2 - 9/2 + y^2 = 9 so

2 ( x + 3/2)^2 + y^2 = 9/2 so

(x+3/2)^2 / (9/4) + y^2 / (9/2) = 1.

This is the equation of an ellipse centered at (-3/2, 0) with semi-axes 3/2 in the x direction and 3 sqrt(2) / 2 in the y direction.

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Self-critique (if necessary):

I am not sure if I am correct. After studying the given solution, I am not sure it is correct as well.

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Self-critique Rating: 1

@& See my note on the last step of your solution. Well done.

The given solution failed to square the coefficient 3 on the left-hand side.

We get

3sqrt(x^2+y^2)=3 + x and square both sides:

9(x^2 + y^2) = x^2 + 6 x + 9, which simplifies to

8 x^2 - 6 x - 9 + 9 y^2 = 0

or

x^2 - 3/4 x + 9 y^2 / 8 = 9/8

Completing the square on x^2 - 3/4 x we get

x ^2 - 3/4 x + (3/8)^2 - (3/8)^2 + 9 y^2 / 8 = 9/8

so

(x - 3/8)^2 + 9 y^2 / 8 = 9/8 + (3/8)^2 = 81 / 64 = (9/8)^2

so that

(x - 3/8)^2 / (9/8)^2 + y^2 / (9/8) = 1

This is the equation of an ellipse centered at (3/8, 0) with semi-axes 9/8 in the x direction and sqrt(9/8) = 3 sqrt(2) / 4 in the y direction.*@

**** Query problem 8.2.10 graph r = 2 sin(`theta).

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Your solution:

We have the polar equation r = 2 sin(theta).

We know that this equation is a circle because it is an equation of the form r = 2a sin (theta) where the circle has radius 1 and center at (0, 1) in rectangular coordinates.

Using symmetry, we find that the graph is symmetric with respect to the line theta = pi/2 so we assign values to theta from -pi/2 to pi/2 in order to plot the polar coordinates as follows:

theta r = 2 sin(theta)

-pi/2 2(-1) = -2

-pi/3 2(-sqrt(3)/2) = is approx. -1.73

-pi/6 2(-1/2) = -1

0 2 (0) = 0

pi/6 2 (1/2) = 1

pi/3 2(sqrt(3)/2) is approx. 1.73

pi/2 2(1) = 2

From the table we have obtained the polar coordinates (-2, -pi/2), (-1.73, -pi/3), (-1, -pi/6), (0, 0), (1, pi/6), (1.73, pi/3), and (2, pi/2). The coordinates are connected to form the graph of the circle.

confidence rating #$&*: 3

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Given Solution:

** When `theta = 0 or `pi, r = 0 and the graph point coincides with the origin.

If `theta = `pi/2, r = 2. If `theta = `pi/4, r = `sqrt(2). So as `theta increases from 0 to `pi/2, r increases from 0 to 2 and the graph follows an arc (actually the arc of a circle) in the first quadrant from the origin to (2, `pi/2).

As `theta moves from `pi/2 to `pi the graph follows an arc in the second quadrant which leads back to the origin.

As `theta goes from `pi to 3 `pi/2, angles in the third quadrant, r becomes negative since sin(`theta) is negative. At 3 `pi / 2, r will be -2 and the point (-2, 3 `pi / 2) coincides with (2, `pi/2). The graph follows the same arc as before in the first quadrant.

As `theta goes from 3 `pi / 2 to 2 `pi, r will remain negative, which places the graph along the same second-quadrant arc as before.

Thus the graph will consist of a closed arc in the upper half-plane, tangent to the x axis at the origin.

This description doesn’t prove that the graph is a circle, but it turns out to be a circle whose radius is 1. The easiest way to prove this is to convert the equation to rectangular coordinates, as follows:

We can multiply both sides by r to get

r ^2 = 2 r sin (theta).

Substituting x^2 + y^2 for r and y for r sin(theta) we have

x ^2 + y ^2 = 2 y

x ^2 + ( y ^2 -2 y ) = 0

x ^ 2 + ( y ^2 - 2 y + 1 - 1 ) = 0

x ^2 + ( y -1 ) ^2 - 1 = 0

x^2 + (y-1)^2 = 1.

This is the standard form of the equation of a circle with center (0, 1) and radius 1. This is the circle described above. **

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Self-critique (if necessary):

OK

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Self-critique Rating: OK

**** Was it possible to use symmetry in any way to obtain your graph,

and if so how did you use it?

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Your solution:

Yes, the graph is symmetric with the line theta = pi/2 and also with horizontal line 1 unit above the pole.

confidence rating #$&*: 3

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Given Solution:

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11:54:29

yes the horizontal symmetry caused us to bound the graph at theta=pi/2 so r

has a maximum height of 2. and its minimum horizontal line is drawn at -2

this maximum and minimum happens alternatley at the intervals of pi/2 and

the negatives.

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11:54:30

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Self-critique (if necessary):

OK

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Self-critique Rating: OK

**** Query problem 8.2.36 graph r=2+4 cos `theta

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11:58:23

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Your solution:

We have the polar equation r = 2 + 4cos(theta).

We know that this equation is a limacon with inner loop because it is an equation of the form r = a + bcos(theta) where a > 0, b > 0, and a < b.

Using symmetry, we find that the graph is symmetric with respect to the polar axis so we assign values to theta from 0 to pi in order to plot the polar coordinates as follows:

theta r = 2 + 4cos(theta)

0 2 + 4(1) = 6

pi/6 2 + 4(sqrt(3)/2) = is approx. 5.46

pi/3 2 + 4(1/2) = 4

pi/2 2 + 4(0) = 2

2pi/3 2 + 4(-1/2) is approx. 2.09

5pi/6 2 + 4(-sqrt(3)/2) is approx. -1.46

pi 2 + 4(-1) = -2

From the table we have obtained the polar coordinates (6, 0), (5.46, pi/6), (4, pi/3), (2, pi/2), (2.09, 2pi/3), (-1.46, 5pi/6), and (-2, pi). The coordinates are connected and reflected across the polar axis to form the graph.

confidence rating #$&*: 3

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Given Solution:

** This graph is symmetric with respect to the pole, since replacing theta by -theta doesn't affect the equation (this is so because cos(-theta) = cos(theta) ).

At theta goes from 0 to pi/2, cos(theta) goes from 1 to 0 so that r will go from 6 to 2.

Then when theta = pi/3 we have cos(theta) = -1/2 so that r = 0. As theta goes from pi/2 to pi/3, then, r goes from 2 to 0. To this point the graph forms half of a heart-shaped figure lying above the x axis.

Between theta = pi/3 and theta = 2 pi / 3 the value of cos(theta) will go from -1/2 to -1 to -1/2, so that r will go from 0 to -2 and back to 0. The corresponding points will move from the origin to the 4th quadrant as theta goes past pi / 3, reaching the point 2 units along the pole when theta = pi then moving into the first quadrant, again reaching the origin when theta = 2 pi/3. These values will form an elongated loop inside the heart-shaped figure.

From theta = 2 pi / 3 thru theta = 3 pi / 2 and on to theta = 2 pi the values of r will go from 0 to 2 then to 6, forming the lower half of the heart-shaped figure. **

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11:58:24

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Self-critique (if necessary):

OK

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Self-critique Rating: OK

See my notes on the one question. The given solution was incorrect; your solution was excellent except that you neglected one step when completing the square.