I need help please!!

Find parametric equations for the path of a particle that moves along the circle x2 + (y − 1)2 = 4 in the manner described. (Ent

QveST [7]

Answer:

$x=2\cos(t)$ and $y=-2\sin(t)+1$

Step-by-step explanation:

$(x-h)^2+(y-k)^2=r^2$ has parametric equations:

$(x-h)=r\cos(t) \text{ and } (y-k)=r\sin(t)$ .

Let's solve these for x and y respectively.

$x-h=r\cos(t)$ can be solved for x by adding h on both sides:

$x=r\cos(t)+h$ .

$y-k=r \sin(t)$ can be solve for y by adding k on both sides:

$y=r\sin(t)+k$ .

We can verify this works by plugging these back in for x and y respectively.

Let's do that:

$(r\cos(t)+h-h)^2+(r\sin(t)+k-k)^2$

$(r\cos(t))^2+(r\sin(t))^2$

$r^2\cos^2(t)+r^2\sin^2(t)$

$r^2(\cos^2(t)+\sin^2(t))$

$r^2(1)$ By a Pythagorean Identity.

$r^2$ which is what we had on the right hand side.

We have confirmed our parametric equations are correct.

Now here your h=0 while your k=1 and r=2.

So we are going to play with these parametric equations:

$x=2\cos(t)$ and $y=2\sin(t)+1$

We want to travel clockwise so we need to put -t and instead of t.

If we were going counterclockwise it would be just the t.

$x=2\cos(-t)$ and $y=2\sin(-t)+1$

Now cosine is even function while sine is an odd function so you could simplify this and say:

$x=2\cos(t)$ and $y=-2\sin(t)+1$ .

We want to find $\theta$ such that

$2\cos(t-\theta_1)=2 \text{ while } -2\sin(t-\theta_2)+1=1$ when t=0.

Let's start with the first equation:

$2\cos(t-\theta_1)=2$

Divide both sides by 2:

$\cos(t-\theta_1)=1$

We wanted to find $\theta_1$ for when $t=0$

$\cos(-\theta_1)=1$

Cosine is an even function:

$\cos(\theta_1)=1$

This happens when $\theta_1=2n\pi$ where n is an integer.

Let's do the second equation:

$-2\sin(t-\theta_2)+1=1$

Subtract 2 on both sides:

$-2\sin(t-\theta_2)=0$

Divide both sides by -2:

$\sin(t-\theta_2)=0$

Recall we are trying to find what $\theta_2$ is when t=0:

$\sin(0-\theta_2)=0$

$\sin(-\theta_2)=0$

Recall sine is an odd function:

$-\sin(\theta_2)=0$

Divide both sides by -1:

$\sin(\theta_2)=0$

$\theta_2=n\pi$

So this means we don't have to shift the cosine parametric equation at all because we can choose n=0 which means $\theta_1=2n\pi=2(0)\pi=0$ .

We also don't have to shift the sine parametric equation either since at n=0, we have $\theta_2=n\pi=0(\pi)=0$ .

So let's see what our equations look like now:

$x=2\cos(t)$ and $y=-2\sin(t)+1$

Let's verify these still work in our original equation:

$x^2+(y-1)^2$

$(2\cos(t))^2+(-2\sin(t))^2$

$2^2\cos^2(t)+(-2)^2\sin^2(t)$

$4\cos^2(t)+4\sin^2(t)$

$4(\cos^2(t)+\sin^2(t))$

$4(1)$

$4$

It still works.

Now let's see if we are being moving around the circle once around for values of t between $0$ and $2\pi$ .

This first table will be the first half of the rotation.

t 0 pi/4 pi/2 3pi/4 pi

x 2 sqrt(2) 0 -sqrt(2) -2

y 1 -sqrt(2)+1 -1 -sqrt(2)+1 1

Ok this is the fist half of the rotation. Are we moving clockwise from (2,1)?

If we are moving clockwise around a circle with radius 2 and center (0,1) starting at (2,1) our x's should be decreasing and our y's should be decreasing at the beginning we should see a 4th of a circle from the point (x,y)=(2,1) and the point (x,y)=(0,-1).

Now after that 4th, the x's will still decrease until we make half a rotation but the y's will increase as you can see from point (x,y)=(0,-1) to (x,y)=(-2,1). We have now made half a rotation around the circle whose center is (0,1) and radius is 2.

Let's look at the other half of the circle:

t pi 5pi/4 3pi/2 7pi/4 2pi

x -2 -sqrt(2) 0 sqrt(2) 2

y 1 sqrt(2)+1 3 sqrt(2)+1 1

So now for the talk half going clockwise we should see the x's increase since we are moving right for them. The y's increase after the half rotation but decrease after the 3/4th rotation.

We also stopped where we ended at the point (2,1).

3 0

3 years ago

What in math what does it mean when I'd a sample process is in control or not?

Aleks04 [339]

I'm not totally sure what you are asking, but maybe this is a question about dependent and independent variables?

If so, these types of variables are used in math and science as measures. In math, independent variables and dependent variables are the "tools" used in an experiment.

Independent variables are the variables that are controlled during the experiment. These variables do not change during the experiment.

Dependent variables are the variables whose values change based on the independent variables. These variables are the "outcome" portion.

I have no idea if this answers your question or not. All I can say is that the "controlled" part of a sample would be the independent variable.

8 0

3 years ago

Anusha needs to find a rectangular container large enough to hold a volume of 130 in.. Which container should sh

Alinara [238K]

Answer:

"a container with the dimensions 3 by 5 by 10 inches"

Step-by-step explanation:

To find volume of rectangular box, we multiply the length, width, and height. Or if given in 3 dimensions form, we multiply all 3.

Lets find volume of each of the answer choices:

a container with the dimensions 2 by 5 by 10 inches = 2 * 5 * 10 = 100

a container with the dimensions 3 by 4 by 10 inches = 3 * 4 * 10 = 120

a container with the dimensions 3 by 5 by 10 inches = 3 * 5 * 10 = 150

a container with the dimensions 4 by 2 by 10 inches = 4 * 2 * 10 = 80

We want a container LARGE ENOUGH to contain a volume of 130 cubic inches. Which of the containers has a volume GREATER than 130?

Yes, only the 3rd option. It is 150 cubic inches in volume.

So, this is the answer:

a container with the dimensions 3 by 5 by 10 inches

4 0

2 years ago

1/8 of a number is 10. What is the number?

OLEGan [10]

For this question you can say:
1/8x = 10
so x is the answer and if u want to find the x you can multiply both sides of the equation by 8 so you will have:
x = 80 and that's the number
if u want to find 1/7 of 21 you should just divide 21 by 7 so you will have 3 and for the last one is same also, so the answer is 5 :)))
i hope this is helpful
have a nice day

6 0

3 years ago

15 POINTS ANSWER ASAP. NO GUESSING

vaieri [72.5K]

9×105=945 and 9×107=963, so 945×963=90,135

5 0

2 years ago

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