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Eva8 [605]
3 years ago
14

Match each of the parametric equations 1-5 with the curve it represents from the list A through E below. 1. r=3 2. r=2sinθ 3. θ=

π4 4. r=4θ 5. r=2cosθ A. A circle of radius 1 with center on the y-axis. B. None of these C. A circle of radius 3 centered at the origin. D. The straight line y=x. E. A circle of radius 1 with center on the x-axis.

Mathematics
1 answer:
ira [324]3 years ago
7 0

Answer:

Step-by-step explanation:

1. r=3

To convert from polar coordinates (r,θ) to rectangular coordinates (x,y), we use the following equations

x=rCosθ

Cosθ=x/r

y=rSinθ

Sinθ=y/r

Also, r²=x²+y²

When r =3,

Then we have

r=3,

x²+y²=3

All points with r = 3 are at

distance 3 from the origin, so r = 3 describes the circle of radius 3, with center at the origin (0,0).

Option C

2. r=2sinθ

When r=2sinθ

To convert from polar coordinates (r,θ) to rectangular coordinates (x,y), we use the following equations

x=rCosθ

Cosθ=x/r

y=rSinθ

Sinθ=y/r

Also, r²=x²+y²

Now, apply the given information

r=2sinθ

Since Sinθ=y/r

r=2y/r

Cross multiply

r²=2y

x²+y²=2y

x²+y²-2y =0

x²+ (y-1)² -1 =0

x²+(y-1)²=1²

Then,

It is a circle with center (0,1) and radius 1.

Because the sine is periodic, we know that we will get the entire curve for values of θ in [0, 2π). As θ runs from 0 to π/2, r increases

from 0 to 2. Then as θ continues to π, r decreases again to 0. When θ runs from π to

2π, r is negative, and it is not hard to see that the first part of the curve is simply traced

out again, so in fact we get the whole curve for values of θ in [0, π). Thus, check attachment for curve, Now, this suggests that the curve could possibly be a circle,

and if it is, it would have to be the circle x² + (y − 1)² = 1. Having made this guess, we can easily check it. First we substitute for x and y to get (r cos θ)² + (r sin θ − 1)² = 1;

expanding and simplifying does indeed turn this into r = 2 sin θ.

Option E

A circle of radius 1 with center on the x-axis

3. θ=π/4

This is a point on the circle

p=(r, θ)

So, r=0

x = rCosπ/4=rCos45

x=r√2 /2

and

y = rsin(π/4) = rSin45

y=r√2 /2

This makes it very

easy to convert equations from rectangular to polar coordinates.

x²+y²= (r√2/2)²+(r√2/2)²

x²+y²= 2r²/4 + 2r²/4

x²+y²= r²

Since r=0

x²+y²= 0

x²=y²

Therefore, x=y

Option D

The straight line y=x

4. r=4θ

Here the distance from the origin

exactly matches the angle, so a bit of thought makes it clear that when θ ≥ 0 we get the spiral of Archimedes ( check attachment ) When θ < 0, r is also negative, and so the full graph is the right hand picture in the figure. Check attachment

The correct option is B

None of the above

5. r=2Cosθ

To convert from polar coordinates (r,θ) to rectangular coordinates (x,y), we use the following equations

x=rCosθ

Cosθ=x/r

y=rSinθ

Sinθ=y/r

Also, r²=x²+y²

Now, apply the given information

r=2cosθ

Since cosθ=x/r

r=2x/r

Cross multiply

r²=2x

x²+y²=2x

x²+y²-2x=0

(x-1)² + y² -1 =0

(x-1)² + y² = 1²

Then,

It is a circle with center (1,0) and radius 1.

A circle of radius 1 with center on the y-axis

Option A is correct

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If one atom of carbon weighs 1.99 × 10^−22 g and one atom of hydrogen weighs 1.67 × 10^−27 g, which element weighs more? Explain
garik1379 [7]

Answer:

The atom of carbon weighs more

Step-by-step explanation:

we have

<em>Atom of carbon weighs</em>

1.99*10^{-22}\ g=1.99*10^{-22}(\frac{10^{5}}{10^{5}})=199,000*10^{-22}(\frac{1}{10^{5}})=199,000*10^{-27}\ g

<em>Atom of hydrogen weighs</em>

1.67*10^{-27}\ g

Compare

199,000*10^{-27}\ g > 1.67*10^{-27}\ g

therefore

The atom of carbon weighs more

8 0
3 years ago
The circle below is centred at point (2,-1) and has a radius of length 3 what is its equation ?
NeTakaya

Answer:

Option C

Step-by-step explanation:

The standard form of equation of a cirle is:

(x-h)^2+ (y-k)^2=r^2

In the given question as the point is given and the radius of circle is given:

So,

(h,k)=(2,-1)

and

r=3

Here,

h=2

k= -1

Putting the values of h,k and r in standard form

(x-2)^2+ (y-(-1))^2=(3)^2

(x-2)^2+ (y+1)^2=(3)^2

So the equation of circle is:

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

Option C is the correct answer ..

7 0
3 years ago
A rectangular prism has a length that is 3 inches shorter than the height and a width that is 5 inches longer than the height. F
LUCKY_DIMON [66]

An equation for the volume of the prism as a function of the height is Volume = h³ + 2h² - 15h.

  • We are given a rectangular prism.
  • A rectangular prism is no different than a cuboid.
  • Let the height of the rectangular prism be "h".
  • The length of the rectangular prism is "h-3".
  • The width of the rectangular prism is "h+5".
  • The volume of the rectangular prism is the same as that of the cuboid.
  • The volume of the rectangular prism is the product of its length, its width, and its height.
  • The volume of the rectangular prism is (h - 3)*(h + 5)*h.
  • An equation for the volume of the prism as a function of the height is :
  • Volume = (h - 3)*(h + 5)*h
  • Volume = (h² + 2h - 15)*h
  • Volume = h³ + 2h² - 15h

To learn more about prism, visit :

brainly.com/question/12649592

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Answer:

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Step-by-step explanation:

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