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nataly862011 [7]
2 years ago
13

Find the centroid of a semi circle using polar coordinates and a double integral. Given a radius of r, and theta being between 0

and pi.
Mathematics
1 answer:
Fittoniya [83]2 years ago
4 0
The area for a semi circle in Cartesian coordinate system with the flat edge against the x-axis is:A=∫0r2−x2√∫−rrdxdy
Bounding limits for x are from -r to r. And for y, this comes from the equation for a circle:x2+y2=r2
Solve for y:y=±r2−x2−−−−−−√
But since we are only dealing with a semi circle, we can always look at the positive answer. 
To convert to a polar coordinate system, we used some equations you have probably seen:x=rcosθy=rsinθdxdy=rdrdθ
The substitute to convert to polar coordinate system:A=∫0π∫0rrdrdθSolve this for the area.A=πr2/2

To find the x coordinate of the centroid:x=1/A∫∫xdxdyorx=1/A∫0π∫0rrcosθ(rdrdθ)Solve and you will find x = 0 which intuitively makes sense.
Similar for the y coordinate of the centroid:y=1/A∫∫ydxdyory=1/A∫0π∫0rrsinθ(rdrdθ)Solve and you will find y=4r/3π
This means the centroid of a semicircle is at: (0,4r/3π)
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Step-by-step explanation:

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Option C.

Step-by-step explanation:

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The quality control manager of a chemical company randomly sampled twenty 100-pound bags of fertilizer to estimate the variance
lisabon 2012 [21]

Answer:

The 95% confidence interval for the variance in the pounds of impurities would be 3.829 \leq \sigma^2 \leq 14.121.

Step-by-step explanation:

1) Data given and notation

s^2 =6.62 represent the sample variance

s=2.573 represent the sample standard deviation

\bar x represent the sample mean

n=20 the sample size

Confidence=95% or 0.95

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population mean or variance lies between an upper and lower interval".  

The margin of error is the range of values below and above the sample statistic in a confidence interval.

The Chi square distribution is the distribution of the sum of squared standard normal deviates .

2) Calculating the confidence interval

The confidence interval for the population variance is given by the following formula:

\frac{(n-1)s^2}{\chi^2_{\alpha/2}} \leq \sigma^2 \frac{(n-1)s^2}{\chi^2_{1-\alpha/2}}

On this case the sample variance is given and for the sample deviation is just the square root of the sample variance.

The next step would be calculate the critical values. First we need to calculate the degrees of freedom given by:

df=n-1=20-1=19

Since the Confidence is 0.95 or 95%, the value of \alpha=0.05 and \alpha/2 =0.025, and we can use excel, a calculator or a table to find the critical values.  

The excel commands would be: "=CHISQ.INV(0.025,19)" "=CHISQ.INV(0.975,19)". so for this case the critical values are:

\chi^2_{\alpha/2}=32.852

\chi^2_{1- \alpha/2}=8.907

And replacing into the formula for the interval we got:

\frac{(19)(6.62)}{32.852} \leq \sigma^2 \frac{(19)(6.62)}{8.907}

3.829 \leq \sigma^2 \leq 14.121

So the 95% confidence interval for the variance in the pounds of impurities would be 3.829 \leq \sigma^2 \leq 14.121.

4 0
3 years ago
A Principal needs 2 chaperones for every 25 students at the school dance how many chaperones does he need if 225 students are ex
Makovka662 [10]

Answer:

18 chaperones

Step-by-step explanation:

1. How many times does 25 go into 225

2. Whatever that number is multiply it by two

3. That is how you get the number of chaperones

4. THE END

5 0
3 years ago
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