To plot: Thepoints which has polar coordinate displaystyle{left({2},frac{{{7}pi}}{{4}}right)} also two alternaitve sets for the same.

To plot: Thepoints which has polar coordinate displaystyle{left({2},frac{{{7}pi}}{{4}}right)} also two alternaitve sets for the same.

Question
Alternate coordinate systems
asked 2021-03-02
To plot: Thepoints which has polar coordinate \(\displaystyle{\left({2},\frac{{{7}\pi}}{{4}}\right)}\) also two alternaitve sets for the same.

Answers (1)

2021-03-03
Let the point \(\displaystyle{\left({2},\frac{{{7}\pi}}{{4}}\right)}\) be denoted by P.
Consider the point \(\displaystyle{\left({2},\frac{{{7}\pi}}{{4}}\right)}\), it is 2 units from the Origin O and line OP makes an angle of \(\displaystyle\frac{{{7}\pi}}{{4}}\) with the positive direction of the x-axis.
Add pi to the angle and change cardinal coordinate to -2 to get one alternate solution.
Therefore, the point P \(\displaystyle{\left({2},\frac{{{7}\pi}}{{4}}\right)}\) can also be represented by \(\displaystyle{\left(-{2},\frac{{{7}\pi}}{{4}}+\pi\right)}={\left(-{2},\frac{{{11}\pi}}{{4}}\right)}.\)
Change the radial coordinate to -2 and subtract \(\displaystyle{3}\pi\) from the angle to obtain the other alternate solution.
Thus, the point \(\displaystyle{P}{\left({2},\frac{{{7}\pi}}{{4}}\right)}\) can be given by \(\displaystyle{\left(-{2},\frac{{{7}\pi}}{{4}}-{3}\pi\right)}\)
\(\displaystyle={\left(-{2},-\frac{{{19}\pi}}{{4}}\right)}.\)
Use online graphing calculator and plot the points of P as shown below in Figure 1.
image
From the Figure 1 it can be noted that alternate representations are plotted on the same location.
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Relevant Questions

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Case: Dr. Jung’s Diamonds Selection
With Christmas coming, Dr. Jung became interested in buying diamonds for his wife. After perusing the Web, he learned about the “4Cs” of diamonds: cut, color, clarity, and carat. He knew his wife wanted round-cut earrings mounted in white gold settings, so he immediately narrowed his focus to evaluating color, clarity, and carat for that style earring.
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2) Let X1, X2, and X3 represent diamond color, clarity, and carats, respectively. If Dr. Jung wanted to build a linear regression model to estimate earring prices using these variables, which variables would you recommend that he use? Why?
3) Suppose Dr. Jung decides to use clarity (X2) and carats (X3) as independent variables in a regression model to predict earring prices. What is the estimated regression equation? What is the value of the R2 and adjusted-R2 statistics?
4) Use the regression equation identified in the previous question to create estimated prices for each of the earring sets in Dr. Jung’s sample. Which sets of earrings appear to be overpriced and which appear to be bargains? Based on this analysis, which set of earrings would you suggest that Dr. Jung purchase?
5) Dr. Jung now remembers that it sometimes helps to perform a square root transformation on the dependent variable in a regression problem. Modify your spreadsheet to include a new dependent variable that is the square root on the earring prices (use Excel’s SQRT( ) function). If Dr. Jung wanted to build a linear regression model to estimate the square root of earring prices using the same independent variables as before, which variables would you recommend that he use? Why?
1
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An automobile tire manufacturer collected the data in the table relating tire pressure x​ (in pounds per square​ inch) and mileage​ (in thousands of​ miles). A mathematical model for the data is given by
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\(\begin{array}{|c|c|} \hline x & Mileage \\ \hline 28 & 45 \\ \hline 30 & 51\\ \hline 32 & 56\\ \hline 34 & 50\\ \hline 36 & 46\\ \hline \end{array}\)
​(A) Complete the table below.
\(\begin{array}{|c|c|} \hline x & Mileage & f(x) \\ \hline 28 & 45 \\ \hline 30 & 51\\ \hline 32 & 56\\ \hline 34 & 50\\ \hline 36 & 46\\ \hline \end{array}\)
​(Round to one decimal place as​ needed.)
\(A. 20602060xf(x)\)
A coordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2. Data points are plotted at (28,45), (30,51), (32,56), (34,50), and (36,46). A parabola opens downward and passes through the points (28,45.7), (30,52.4), (32,54.7), (34,52.6), and (36,46.0). All points are approximate.
\(B. 20602060xf(x)\)
Acoordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2.
Data points are plotted at (43,30), (45,36), (47,41), (49,35), and (51,31). A parabola opens downward and passes through the points (43,30.7), (45,37.4), (47,39.7), (49,37.6), and (51,31). All points are approximate.
\(C. 20602060xf(x)\)
A coordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2. Data points are plotted at (43,45), (45,51), (47,56), (49,50), and (51,46). A parabola opens downward and passes through the points (43,45.7), (45,52.4), (47,54.7), (49,52.6), and (51,46.0). All points are approximate.
\(D.20602060xf(x)\)
A coordinate system has a horizontal x-axis labeled from 20 to 60 in increments of 2 and a vertical y-axis labeled from 20 to 60 in increments of 2. Data points are plotted at (28,30), (30,36), (32,41), (34,35), and (36,31). A parabola opens downward and passes through the points (28,30.7), (30,37.4), (32,39.7), (34,37.6), and (36,31). All points are approximate.
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\(\displaystyle​\frac{{{l}{b}{s}}}{{{s}{q}}}\in.\) and for 35
\(\displaystyle​\frac{{{l}{b}{s}}}{{{s}{q}}}\in.\)
The mileage for the tire pressure \(\displaystyle{29}\frac{{{l}{b}{s}}}{{{s}{q}}}\in.\) is
The mileage for the tire pressure \(\displaystyle{35}\frac{{{l}{b}{s}}}{{{s}{q}}}\) in. is
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Prove that by setting these two expressions equal to another, the result is an identity:
\(\displaystyle{d}={{\cos}^{ -{{1}}}{\left( \sin{{\left({L}{T}_{{1}}\right)}} \sin{{\left({L}{T}_{{2}}\right)}}+ \cos{{\left({L}{T}_{{1}}\right)}} \cos{{\left({L}{T}_{{2}}\right)}} \cos{{\left({{\ln}_{{1}}-}{\ln}_{{2}}\right)}}\right)}}\)
\(\displaystyle{d}={2}{{\sin}^{ -{{1}}}{\left(\sqrt{{{{\sin}^{2}{\left(\frac{{{L}{T}_{{1}}-{L}{T}_{{2}}}}{{2}}\right)}}+ \cos{{\left({L}{T}_{{1}}\right)}} \cos{{\left({L}{T}_{{2}}\right)}}{{\sin}^{2}{\left(\frac{{{{\ln}_{{1}}-}{\ln}_{{2}}}}{{2}}\right)}}}}\right)}}\)
asked 2021-05-12
4.7 A multiprocessor with eight processors has 20attached tape drives. There is a large number of jobs submitted tothe system that each require a maximum of four tape drives tocomplete execution. Assume that each job starts running with onlythree tape drives for a long period before requiring the fourthtape drive for a short period toward the end of its operation. Alsoassume an endless supply of such jobs.
a) Assume the scheduler in the OS will not start a job unlessthere are four tape drives available. When a job is started, fourdrives are assigned immediately and are not released until the jobfinishes. What is the maximum number of jobs that can be inprogress at once? What is the maximum and minimum number of tapedrives that may be left idle as a result of this policy?
b) Suggest an alternative policy to improve tape driveutilization and at the same time avoid system deadlock. What is themaximum number of jobs that can be in progress at once? What arethe bounds on the number of idling tape drives?
asked 2020-11-22
To find: The equivalent polar equation for the given rectangular-coordinate equation.
Given:
\(\displaystyle\ {x}=\ {r}{\cos{\theta}}\)
\(\displaystyle\ {y}=\ {r}{\sin{\theta}}\)
b. From rectangular to polar:
\(\displaystyle{r}=\pm\sqrt{{{x}^{{{2}}}\ +\ {y}^{{{2}}}}}\)
\(\displaystyle{\cos{\theta}}={\frac{{{x}}}{{{r}}}},{\sin{\theta}}={\frac{{{y}}}{{{r}}}},{\tan{\theta}}={\frac{{{x}}}{{{y}}}}\)
Calculation:
Given: equation in rectangular-coordinate is \(\displaystyle{y}={x}\).
Converting into equivalent polar equation -
\(\displaystyle{y}={x}\)
Put \(\displaystyle{x}={r}{\cos{\theta}},\ {y}={r}{\sin{\theta}},\)
\(\displaystyle\Rightarrow\ {r}{\sin{\theta}}={r}{\cos{\theta}}\)
\(\displaystyle\Rightarrow\ {\frac{{{\sin{\theta}}}}{{{\cos{\theta}}}}}={1}\)
\(\displaystyle\Rightarrow\ {\tan{\theta}}={1}\)
Thus, desired equivalent polar equation would be \(\displaystyle\theta={1}\)
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