Suppose the manufacturer of widgets has developed the following table showing the highest price p, in dollars, of a widget at which N widgets can be sold. begin{array}{|c|c|} hline Number N & Price p hline 200 & 53.00 hline 250 & 52.50hline 300 & 52.00 hline 350 & 51.50 hline end{array} (a) Find a formula for p in terms of N modeling the data in the table. displaystyle{p}= (b) Use a formula to express the total monthly revenue R, in dollars, of this manufacturer in a month as a function of the number N of widgets produced in a month. displaystyle{R}= Is R a linear function of N? (c) On the basis of the tables in this exercise and using cost, displaystyle{C}={35}{N}+{900}, use a formula to express the monthly profit P, in dollars, of this manufacturer as a function of the number of w

Question
Modeling data distributions
asked 2021-03-05
Suppose the manufacturer of widgets has developed the following table showing the highest price p, in dollars, of a widget at which N widgets can be sold.
\(\begin{array}{|c|c|} \hline Number\ N & Price\ p\\ \hline 200 & 53.00\\ \hline 250 & 52.50\\\hline 300 & 52.00\\ \hline 350 & 51.50\\ \hline \end{array}\)
(a) Find a formula for p in terms of N modeling the data in the table.
\(\displaystyle{p}=\)
(b) Use a formula to express the total monthly revenue R, in dollars, of this manufacturer in a month as a function of the number N of widgets produced in a month.
\(\displaystyle{R}=\)
Is R a linear function of N?
(c) On the basis of the tables in this exercise and using cost, \(\displaystyle{C}={35}{N}+{900}\), use a formula to express the monthly profit P, in dollars, of this manufacturer as a function of the number of widgets produced in a month.
\(\displaystyle{P}=\)
(d) Is P a linear function of N?

Answers (1)

2021-03-06
As per bartleby guidelines for the more than 3 subparts only first three are to be answered please upload the others separately.
(a)
Consider the two points from table (a) as (200 53.00) and (250 52.50).
Then, the equation is,
\(\displaystyle{p}-{p}_{{1}}=\frac{{{p}_{{2}}-{p}_{{1}}}}{{{N}_{{2}}-{N}_{{1}}}}{\left({N}-{N}_{{1}}\right)}\)
\(\displaystyle{p}-{53.00}=\frac{{{52.50}-{53.00}}}{{{250}-{200}}}{\left({N}-{200}\right)}\)
\(\displaystyle{p}-{53}=\frac{{-{0.50}}}{{50}}{\left({N}-{200}\right)}\)
\(\displaystyle{p}-{53}=-{0.01}{N}+{2}\)
\(\displaystyle{p}=-{0.01}{N}+{55}\)
Thus, the equation \(\displaystyle{p}=-{0.01}{N}+{55}\)
(b)
Consider the formula to express the total monthly revenue R, in dollars, of this manufacturer in a month as a function of the number N of widgets produced in a month.
\(\displaystyle{R}={\left(-{0.01}{N}+{55}\right)}{N}\)
\(\displaystyle=-{0.01}{N}^{2}+{55}{N}\)
Thus, the formula is \(\displaystyle{R}=-{0.01}{N}^{2}+{55}{N}.\)
Here, R is not the linear function of N.
(c)
The formula for the monthly profit is,
\(\displaystyle{P}=-{0.01}{N}^{2}+{55}{N}-{35}{N}-{900}\)
\(\displaystyle=-{0.01}{N}^{2}+{20}{N}-{900}\)
Here, P is not the linear function of N.
0

Relevant Questions

asked 2021-03-11
Let's say the widget maker has developed the following table that shows the highest dollar price p. widget where you can sell N widgets. Number N Price p \(200 53.00\)
\(250 52.50\)
\(300 52.00\)
\(35051.50\) (a) Find a formula for pin terms of N modeling the data in the table. (b) Use a formula to express the total monthly revenue R, in dollars, of this manufacturer in month as a function of the number N of widgets produced in a month. \(R=\) Is Ra linear function of N? (c) On the basis of the tables in this exercise and using cost, \(C= 35N + 900\), use a formula to express the monthly profit P, in dollars, of this manufacturer asa function of the number of widgets produced in a month \(p=\) (d) Is Pa linear function of N2 e. Explain how you would find breakeven. What does breakeven represent?
asked 2021-02-25
In general, the highest price p per unit of an item at which a manufacturer can sell N items is not constant but is, rather, a function of N. Suppose the manufacturer of widgets has developed the following table showing the highest price p, in dollars, of a widget at which N widgets can be sold.
\(\begin{array}{|c|c|} \hline Number\ N & Price\ p\\ \hline 250 & 52.50\\ \hline300 & 52.00\\\hline 350 & 51.50\\ \hline 400 & 51.00\\ \hline \end{array}\)
(a) Find a formula for p in terms of N modeling the data in the table.
\(\displaystyle{p}=\)
(b) Use a formula to express the total monthly revenue R, in dollars, of this manufacturer in a month as a function of the number N of widgets produced in a month.
\(\displaystyle{R}=\)
asked 2020-12-28
In general, the highest price p per unit of an item at which a manufacturer can sell N items is not constant but is, rather, a function of N. Suppose the manufacturer of widgets has developed the following table showing the highest price p, in dollars, of a widget at which N widgets can be sold.
\(\begin{array}{|c|c|} \hline Number\ N & Price\ p\\ \hline 250 & 41.50\\ \hline300 & 40.80\\\hline 350 & 40.10\\ \hline 400 & 39.40\\ \hline \end{array}\)
Find a formula for p in terms of N modeling the data in the table.
asked 2021-03-11
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
\(\displaystyle​ f{{\left({x}\right)}}=-{0.554}{x}^{2}+{35.5}{x}-{514}.\)
\(\begin{array}{|c|c|} \hline x & Mileage \\ \hline 28 & 45 \\ \hline 30 & 51\\ \hline 32 & 56\\ \hline 34 & 50\\ \hline 36 & 46\\ \hline \end{array}\)
​(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.
\(D.20602060xf(x)\)
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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
(Round to two decimal places as​ needed.)
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A. As tire pressure​ increases, mileage decreases to a minimum at a certain tire​ pressure, then begins to increase.
B. As tire pressure​ increases, mileage decreases.
C. As tire pressure​ increases, mileage increases to a maximum at a certain tire​ pressure, then begins to decrease.
D. As tire pressure​ increases, mileage increases.
asked 2020-10-23
A random sample of \(\displaystyle{n}_{{1}}={16}\) communities in western Kansas gave the following information for people under 25 years of age.
\(\displaystyle{X}_{{1}}:\) Rate of hay fever per 1000 population for people under 25
\(\begin{array}{|c|c|} \hline 97 & 91 & 121 & 129 & 94 & 123 & 112 &93\\ \hline 125 & 95 & 125 & 117 & 97 & 122 & 127 & 88 \\ \hline \end{array}\)
A random sample of \(\displaystyle{n}_{{2}}={14}\) regions in western Kansas gave the following information for people over 50 years old.
\(\displaystyle{X}_{{2}}:\) Rate of hay fever per 1000 population for people over 50
\(\begin{array}{|c|c|} \hline 94 & 109 & 99 & 95 & 113 & 88 & 110\\ \hline 79 & 115 & 100 & 89 & 114 & 85 & 96\\ \hline \end{array}\)
(i) Use a calculator to calculate \(\displaystyle\overline{{x}}_{{1}},{s}_{{1}},\overline{{x}}_{{2}},{\quad\text{and}\quad}{s}_{{2}}.\) (Round your answers to two decimal places.)
(ii) Assume that the hay fever rate in each age group has an approximately normal distribution. Do the data indicate that the age group over 50 has a lower rate of hay fever? Use \(\displaystyle\alpha={0.05}.\)
(a) What is the level of significance?
State the null and alternate hypotheses.
\(\displaystyle{H}_{{0}}:\mu_{{1}}=\mu_{{2}},{H}_{{1}}:\mu_{{1}}<\mu_{{2}}\)
\(\displaystyle{H}_{{0}}:\mu_{{1}}=\mu_{{2}},{H}_{{1}}:\mu_{{1}}>\mu_{{2}}\)
\(\displaystyle{H}_{{0}}:\mu_{{1}}=\mu_{{2}},{H}_{{1}}:\mu_{{1}}\ne\mu_{{2}}\)
\(\displaystyle{H}_{{0}}:\mu_{{1}}>\mu_{{2}},{H}_{{1}}:\mu_{{1}}=\mu_{{12}}\)
(b) What sampling distribution will you use? What assumptions are you making?
The standard normal. We assume that both population distributions are approximately normal with known standard deviations.
The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations,
The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations,
The Student's t. We assume that both population distributions are approximately normal with known standard deviations,
What is the value of the sample test statistic? (Test the difference \(\displaystyle\mu_{{1}}-\mu_{{2}}\). Round your answer to three decimalplaces.)
What is the value of the sample test statistic? (Test the difference \(\displaystyle\mu_{{1}}-\mu_{{2}}\). Round your answer to three decimal places.)
(c) Find (or estimate) the P-value.
P-value \(\displaystyle>{0.250}\)
\(\displaystyle{0.125}<{P}-\text{value}<{0},{250}\)
\(\displaystyle{0},{050}<{P}-\text{value}<{0},{125}\)
\(\displaystyle{0},{025}<{P}-\text{value}<{0},{050}\)
\(\displaystyle{0},{005}<{P}-\text{value}<{0},{025}\)
P-value \(\displaystyle<{0.005}\)
Sketch the sampling distribution and show the area corresponding to the P-value.
P.vaiue Pevgiue
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asked 2020-10-23
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asked 2020-11-07
1)A rewiew of voted registration record in a small town yielded the dollowing data of the number of males and females registered as Democrat, Republican, or some other affilation: \(\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{c}\right\rbrace}{G}{e}{n}{d}{e}{r}\backslash{h}{l}\in{e}{A}{f}{f}{i}{l}{a}{t}{i}{o}{n}&{M}{a}\le&{F}{e}{m}{a}\le\backslash{h}{l}\in{e}{D}{e}{m}{o}{c}{r}{a}{t}&{300}&{600}\backslash{R}{e}{p}{u}{b}{l}{i}{c}{a}{n}&{500}&{300}\backslash{O}{t}{h}{e}{r}&{200}&{100}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}\) What proportion of all voters is male and registered as a Democrat? 2)A survey was conducted invocted involving 303 subject concerning their preferences with respect to the size of car thay would consider purchasing. The following table shows the count of the responses by gender of the respondents: \(\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{c}\right\rbrace}{S}{i}{z}{e}\ {o}{f}\ {C}{a}{r}\backslash{h}{l}\in{e}{G}{e}{n}{d}{e}{r}&{S}{m}{a}{l}{l}&{M}{e}{d}{i}{u}{m}&{l}{a}{n}\ge&{T}{o}{t}{a}{l}\backslash{h}{l}\in{e}{F}{e}{m}{a}\le&{58}&{63}&{17}&{138}\backslash{M}{a}\le&{79}&{61}&{25}&{165}\backslash{T}{o}{t}{a}{l}&{137}&{124}&{42}&{303}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}\) the data are to be summarized by constructing marginal distributions. In the marginal distributio for car size, the entry for mediums car is ?
asked 2020-12-02
Gastroenterology
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If we do not want to assume normality for these distributions, then what statistical procedure can be used to compare the three groups?
Perform the test mentioned in Problem 12.42 and report a p-value. How do your results compare with a parametric analysis of the data?
Relationship between protein concentration \((mg/mL)\) of duodenal secretions to pancreatic function as measured by trypsin secretion:
\(\left[U/\left(k\ \frac{g}{h}r\right)\right]\)
Tapsin secreton [UGA]
\(\leq\ 50\)
\(\begin{array}{|c|c|}\hline \text{Subject number} & \text{Protetion concentration} \\ \hline 1 & 1.7 \\ \hline 2 & 2.0 \\ \hline 3 & 2.0 \\ \hline 4 & 2.2 \\ \hline 5 & 4.0 \\ \hline 6 & 4.0 \\ \hline 7 & 5.0 \\ \hline 8 & 6.7 \\ \hline 9 & 7.8 \\ \hline \end{array}\)
\(51\ -\ 1000\)
\(\begin{array}{|c|c|}\hline \text{Subject number} & \text{Protetion concentration} \\ \hline 1 & 1.4 \\ \hline 2 & 2.4 \\ \hline 3 & 2.4 \\ \hline 4 & 3.3 \\ \hline 5 & 4.4 \\ \hline 6 & 4.7 \\ \hline 7 & 6.7 \\ \hline 8 & 7.9 \\ \hline 9 & 9.5 \\ \hline 10 & 11.7 \\ \hline \end{array}\)
\(>\ 1000\)
\(\begin{array}{|c|c|}\hline \text{Subject number} & \text{Protetion concentration} \\ \hline 1 & 2.9 \\ \hline 2 & 3.8 \\ \hline 3 & 4.4 \\ \hline 4 & 4.7 \\ \hline 5 & 5.5 \\ \hline 6 & 5.6 \\ \hline 7 & 7.4 \\ \hline 8 & 9.4 \\ \hline 9 & 10.3 \\ \hline \end{array}\)
asked 2020-11-23
The accompanying two-way table was constructed using data in the article “Television Viewing and Physical Fitness in Adults” (Research Quarterly for Exercise and Sport, 1990: 315–320). The author hoped to determine whether time spent watching television is associated with cardiovascular fitness. Subjects were asked about their television-viewing habits and were classified as physically fit if they scored in the excellent or very good category on a step test. We include MINITAB output from a chi-squared analysis. The four TV groups corresponded to different amounts of time per day spent watching TV (0, 1–2, 3–4, or 5 or more hours). The 168 individuals represented in the first column were those judged physically fit. Expected counts appear below observed counts, and MINITAB displays the contribution to \(\displaystyle{x}^{{{2}}}\) from each cell.
State and test the appropriate hypotheses using \(\displaystyle\alpha={0.05}\)
\(\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{h}{l}\in{e}&{a}\mp,\ {1}&{a}\mp,\ {2}&{a}\mp,\ {T}{o}{t}{a}{l}\backslash{h}{l}\in{e}{1}&{a}\mp,\ {35}&{a}\mp,\ {147}&{a}\mp,\ {182}\backslash{h}{l}\in{e}&{a}\mp,\ {25.48}&{a}\mp,\ {156.52}&{a}\mp,\backslash{h}{l}\in{e}{2}&{a}\mp,\ {101}&{a}\mp,\ {629}&{a}\mp,\ {730}\backslash{h}{l}\in{e}&{a}\mp,\ {102.20}&{a}\mp,\ {627.80}&{a}\mp,\backslash{h}{l}\in{e}{3}&{a}\mp,\ {28}&{a}\mp,\ {222}&{a}\mp,\ {250}\backslash{h}{l}\in{e}&{a}\mp,\ {35.00}&{a}\mp,\ {215.00}&{a}\mp,\backslash{h}{l}\in{e}{4}&{a}\mp,\ {4}&{a}\mp,\ {34}&{a}\mp,\ {38}\backslash{h}{l}\in{e}&{a}\mp,\ {5.32}&{a}\mp,\ {32.68}&{a}\mp,\backslash{h}{l}\in{e}{T}{o}{t}{a}{l}&{a}\mp,\ {168}&{a}\mp,\ {1032}&{a}\mp,\ {1200}\backslash{h}{l}\in{e}\)
\(\displaystyle{C}{h}{i}{s}{q}={a}\mp,\ {3.557}\ +\ {0.579}\ +\ {a}\mp,\ {0.014}\ +\ {0.002}\ +\ {a}\mp,\ {1.400}\ +\ {0.228}\ +\ {a}\mp,\ {0.328}\ +\ {0.053}={6.161}\)
\(\displaystyle{d}{f}={3}\)
asked 2020-11-26
Two drugs, Abraxane and Taxol, are both cancer treatments, yet have differing rates at which they leave a patient’s system. Using terminology from pharmacology, Abraxane leaves the system by so-called “first-order elimination”, which means that the concentration decreases at a constant percentage rate for each unit of time that passes. Taxol leaves the system by “zero-order elimination”, which means that the concentration decreases by a constant amount for each unit of time that passes.
(a) As soon as the infusion of Taxol is completed, the drug concentration in a patient’s blood is 1000 nanograms per milliliter \(\displaystyle{\left(\frac{{{n}{g}}}{{{m}{l}}}\right)}.\) 12 hours later there is \(\displaystyle{50}\frac{{{n}{g}}}{{{m}{l}}}\) left in the patient’s system. Use the data to construct an appropriate formula modeling the blood concentration of Taxol as a function of time after the infusion is completed.
(b) As soon as the infusion of Abraxane is completed, the drug concentration in a patient’s blood is 1000 nanograms per milliliter \(\displaystyle{\left(\frac{{{n}{g}}}{{{m}{l}}}\right)}\). 24 hours later there is \(\displaystyle{50}\frac{{{n}{g}}}{{{m}{l}}}\) left in the patient’s system. Use the data to construct an appropriate formula modeling the blood concentration of Abraxane as a function of time after the infusion is completed.
(c) Find the long-term behavior of the function from part (b). Is this behavior meaningful in the context of the model?
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