Applications using double integrals: A lamina occupies the part of the disk x^2 + y^2 <= 1 in the first quadrant. Use polar coordinates to find the center of mass of the lamina if the density at any point is proportional to the square of its distance from the origin.
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2021-02-04
Relevant Questions
asked 2020-10-20
Applications of double integrals:
A lamina occupies the part of the disk \(\displaystyle{x}^{{2}}+{y}^{{2}}\le{4}\) in the first quadrant. Find the center of mass of the lamina if the density at any point is proportional to the square of its distance from the origin.
A lamina occupies the part of the disk \(\displaystyle{x}^{{2}}+{y}^{{2}}\le{4}\) in the first quadrant. Find the center of mass of the lamina if the density at any point is proportional to the square of its distance from the origin.
asked 2021-02-27
Applications of double integrals:
A lamina occupies the part of the disk \(\displaystyle{x}^{{2}}+{y}^{{2}}\le{64}\) in the first quadrant. Find its center of mass if the density at any point is proportional to its distance from the x-axis.
A lamina occupies the part of the disk \(\displaystyle{x}^{{2}}+{y}^{{2}}\le{64}\) in the first quadrant. Find its center of mass if the density at any point is proportional to its distance from the x-axis.
asked 2020-10-23
1. Find each of the requested values for a population with a mean of \(? = 40\), and a
standard deviation of \(? = 8\)
A. What is the z-score corresponding to \(X = 52?\)
B. What is the X value corresponding to \(z = - 0.50?\)
C. If all of the scores in the population are transformed into z-scores, what will be the values for the mean and standard deviation for the complete set of z-scores?
D. What is the z-score corresponding to a sample mean of \(M=42\) for a sample of \(n = 4\) scores?
E. What is the z-scores corresponding to a sample mean of \(M= 42\) for a sample of \(n = 6\) scores?
2. True or false:
a. All normal distributions are symmetrical
b. All normal distributions have a mean of 1.0
c. All normal distributions have a standard deviation of 1.0
d. The total area under the curve of all normal distributions is equal to 1
3. Interpret the location, direction, and distance (near or far) of the following zscores: \(a. -2.00 b. 1.25 c. 3.50 d. -0.34\)
4. You are part of a trivia team and have tracked your team’s performance since you started playing, so you know that your scores are normally distributed with \(\mu = 78\) and \(\sigma = 12\). Recently, a new person joined the team, and you think the scores have gotten better. Use hypothesis testing to see if the average score has improved based on the following 8 weeks’ worth of score data: \(82, 74, 62, 68, 79, 94, 90, 81, 80\).
5. You get hired as a server at a local restaurant, and the manager tells you that servers’ tips are $42 on average but vary about \($12 (\mu = 42, \sigma = 12)\). You decide to track your tips to see if you make a different amount, but because this is your first job as a server, you don’t know if you will make more or less in tips. After working 16 shifts, you find that your average nightly amount is $44.50 from tips. Test for a difference between this value and the population mean at the \(\alpha = 0.05\) level of significance.
asked 2020-12-29
A rubber ball of mass m is dropped from a cliff. As theball falls. it is subject to air drag (a resistive force caused bythe air). The drag force on the ball has magnitudebv^{2}, where b is a canstant drag coefficient andv is the instantaneous speed of the ball. The dragcoefficient b is directly proportional to the cross-sectional areaof the ball and the density of the air and does not depend on themass of the ball. As the ball falls, its speedapproaches a constant value called the terminal speed.
a. Write, but do Not solve, a differentialequation for the instantaneous speed v of the ball in terms of timet, the given quantities quantities, and fundamentalconstants.
b. Determine the terminal speed vt interms of the given quantities and fundamental constants.
c. Detemine the energy dissipated by the dragforce during the fall if the ball is released at height h andreaches its reminal speed before hitting the ground, in terms ofthe given quantities and fundamental constants.
a. Write, but do Not solve, a differentialequation for the instantaneous speed v of the ball in terms of timet, the given quantities quantities, and fundamentalconstants.
b. Determine the terminal speed vt interms of the given quantities and fundamental constants.
c. Detemine the energy dissipated by the dragforce during the fall if the ball is released at height h andreaches its reminal speed before hitting the ground, in terms ofthe given quantities and fundamental constants.
asked 2020-10-28
True or False: In many applications of definite integrals, the integral is used to compute the total amount of a varying quantity.
asked 2020-12-14
Let C be a circle, and let P be a point not on the circle. Prove that the maximum and minimum distances from P to a point X on C occur when the line X P goes through the center of C. [Hint: Choose coordinate systems so that C is defined by
\(x2 + y2 = r2\) and P is a point (a,0)
on the x-axis with a \(\neq \pm r,\) use calculus to find the maximum and minimum for the square of the distance. Don’t forget to pay attention to endpoints and places where a derivative might not exist.]
\(x2 + y2 = r2\) and P is a point (a,0)
on the x-axis with a \(\neq \pm r,\) use calculus to find the maximum and minimum for the square of the distance. Don’t forget to pay attention to endpoints and places where a derivative might not exist.]
asked 2021-02-06
At what age do babies learn to crawl? Does it take longer to learn in the winter when babies are often bundled in clothes that restrict their movement? Data were collected from parents who brought their babies into the University of Denver Infant Study Center to participate in one of a number of experiments between 1988 and 1991. Parents reported the birth month and the age at which their child was first able to creep or crawl
a distance of 4 feet within 1 minute. The resulting data were grouped by month of birth: January, May, and September:
\(\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{c}\right\rbrace}&{C}{r}{a}{w}{l}\in{g}\ {a}\ge\backslash{h}{l}\in{e}{B}{i}{r}{t}{h}\ {m}{o}{n}{t}{h}&{M}{e}{a}{n}&{S}{t}.{d}{e}{v}.&{n}\backslash{h}{l}\in{e}{J}{a}\nu{a}{r}{y}&{29.84}&{7.08}&{32}\backslash{M}{a}{y}&{28.58}&{8.07}&{27}\backslash{S}{e}{p}{t}{e}{m}{b}{e}{r}&{33.83}&{6.93}&{38}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}\)
Crawling age is given in weeks. Assume the data represent three independent simple random samples, one from each of the three populations consisting of babies born in that particular month, and that the populations
of crawling ages have Normal distributions.
A partial ANOVA table is given below.
\(\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{c}\right\rbrace}{S}{o}{u}{r}{c}{e}&{S}{u}{m}\ {o}{f}\ \boxempty{s}&{D}{F}&{M}{e}{a}{n}\ \boxempty\ {F}\backslash{h}{l}\in{e}{G}{r}{o}{u}{p}{s}&{505.26}\backslash{E}{r}{r}{\quad\text{or}\quad}&&&{53.45}\backslash{T}{o}{t}{a}{l}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}\)
What are the degrees of freedom for the groups term?
asked 2020-10-28
A wood cube 0.30 m on each side has a density of 700kg/m3 and floats levelly in water. (a) What is the distance from the top of the wood to the water surface? (b) What mass has to be placed on top of the wood so that its top isjust at the water level?
asked 2020-11-08
To determine:
a) The origin \(\displaystyle{\left[\begin{array}{cc} {0}&\ {0}\end{array}\right]}\) is a critical point of the systems.
\(\displaystyle{\left[{\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}={y}\ +\ {x}{\left({x}^{{{2}}}\ +\ {y}^{{{2}}}\right)},\ {\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}=\ -{x}\ +\ {y}{\left({x}^{{{2}}}\ +\ {y}^{{{2}}}\right)}\right]}{\quad\text{and}\quad}{\left[{\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}={y}\ -\ {x}{\left({x}^{{{2}}}\ +\ {y}^{{{2}}}\right)},\ {\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}=\ -{x}\ -\ {y}{\left({x}^{{{2}}}\ +\ {y}^{{{2}}}\right)}\right]}\). Futhermore, it is a center of the corresponding linear system.
b) The systems \(\displaystyle{\left[{\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}={y}\ +\ {x}{\left({x}^{{{2}}}\ +\ {y}^{{{2}}}\right)},\ {\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}=\ -{x}\ +\ {y}{\left({x}^{{{2}}}\ +\ {y}^{{{2}}}\right)}\right]}{\quad\text{and}\quad}{\left[{\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}={y}\ -\ {y}\ +\ {x}{\left({x}^{{{2}}}\ +\ {y}^{{{2}}}\right)},\ {\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}=\ -{x}\ -\ {y}{\left({x}^{{{2}}}\ +\ {y}^{{{2}}}\right)}\right]}\) are almost linear.
c) To prove: \(\displaystyle{\left[{\frac{{{d}{r}}}{{{\left.{d}{t}\right.}}}}\ {<}\ {0}\right]}{\quad\text{and}\quad}{\left[{r}\rightarrow\ {0}\ {a}{s}\ {t}\rightarrow\ \infty\right]},\) hence the critical point for the system \(\displaystyle{\left[{\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}={y}\ -\ {x}{\left({x}^{{{2}}}\ +\ {y}^{{{2}}}\right)},\ {\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}=\ -{x}\ -\ {y}{\left({x}^{{{2}}}\ +\ {y}^{{{2}}}\right)}\right]}\) is asymptotically stable and the solution of the initial value problem for \(\displaystyle{\left[{r}\ {w}{i}{t}{h}\ {r}={r}_{{{0}}}\ {a}{t}\ {t}={0}\right]}\) becomes unbounded as \(\displaystyle{\left[{t}\rightarrow{\frac{{{1}}}{{{2}}}}\ {r}{\frac{{{2}}}{{{0}}}}\right]}\), hence the critical point for the system \(\displaystyle{\left[{\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}={y}\ +\ {x}{\left({x}^{{{2}}}\ +\ {y}^{{{2}}}\right)},\ {\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}=\ -{x}\ +\ {y}{\left({x}^{{{2}}}\ +\ {y}^{{{2}}}\right)}\right]}\) is unstable.
a) The origin \(\displaystyle{\left[\begin{array}{cc} {0}&\ {0}\end{array}\right]}\) is a critical point of the systems.
\(\displaystyle{\left[{\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}={y}\ +\ {x}{\left({x}^{{{2}}}\ +\ {y}^{{{2}}}\right)},\ {\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}=\ -{x}\ +\ {y}{\left({x}^{{{2}}}\ +\ {y}^{{{2}}}\right)}\right]}{\quad\text{and}\quad}{\left[{\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}={y}\ -\ {x}{\left({x}^{{{2}}}\ +\ {y}^{{{2}}}\right)},\ {\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}=\ -{x}\ -\ {y}{\left({x}^{{{2}}}\ +\ {y}^{{{2}}}\right)}\right]}\). Futhermore, it is a center of the corresponding linear system.
b) The systems \(\displaystyle{\left[{\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}={y}\ +\ {x}{\left({x}^{{{2}}}\ +\ {y}^{{{2}}}\right)},\ {\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}=\ -{x}\ +\ {y}{\left({x}^{{{2}}}\ +\ {y}^{{{2}}}\right)}\right]}{\quad\text{and}\quad}{\left[{\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}={y}\ -\ {y}\ +\ {x}{\left({x}^{{{2}}}\ +\ {y}^{{{2}}}\right)},\ {\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}=\ -{x}\ -\ {y}{\left({x}^{{{2}}}\ +\ {y}^{{{2}}}\right)}\right]}\) are almost linear.
c) To prove: \(\displaystyle{\left[{\frac{{{d}{r}}}{{{\left.{d}{t}\right.}}}}\ {<}\ {0}\right]}{\quad\text{and}\quad}{\left[{r}\rightarrow\ {0}\ {a}{s}\ {t}\rightarrow\ \infty\right]},\) hence the critical point for the system \(\displaystyle{\left[{\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}={y}\ -\ {x}{\left({x}^{{{2}}}\ +\ {y}^{{{2}}}\right)},\ {\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}=\ -{x}\ -\ {y}{\left({x}^{{{2}}}\ +\ {y}^{{{2}}}\right)}\right]}\) is asymptotically stable and the solution of the initial value problem for \(\displaystyle{\left[{r}\ {w}{i}{t}{h}\ {r}={r}_{{{0}}}\ {a}{t}\ {t}={0}\right]}\) becomes unbounded as \(\displaystyle{\left[{t}\rightarrow{\frac{{{1}}}{{{2}}}}\ {r}{\frac{{{2}}}{{{0}}}}\right]}\), hence the critical point for the system \(\displaystyle{\left[{\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}={y}\ +\ {x}{\left({x}^{{{2}}}\ +\ {y}^{{{2}}}\right)},\ {\frac{{{\left.{d}{x}\right.}}}{{{\left.{d}{t}\right.}}}}=\ -{x}\ +\ {y}{\left({x}^{{{2}}}\ +\ {y}^{{{2}}}\right)}\right]}\) is unstable.
asked 2020-10-27
Two basketball players are essentially equal in all respects. In particular, by jumping they can raise their centers of mass the same vertical distance, H. The first player,Arabella, wishes to shoot over the second player, Boris, and forthis she needs to be as high above Boris as possible. Arabella Jumps at time t=0, and Boris jumps later, at time \(\displaystyle{t}_{{R}}\)(his reaction time). Assume that Arabella has not yet reached her maximum height when Boris jumps.
Part A.) Find the vertical displacement \(\displaystyle{D}{\left({t}\right)}={h}_{{A}}{\left({t}\right)}-{h}_{{B}}{\left({t}\right)}\), as a function of time for the interval \(\displaystyle{0}{<}{t}{<}{t}_{{R}}\), where \(\displaystyle{h}_{{A}}{\left({t}\right)}\) is the height of the raised hands of Arabella, while \(\displaystyle{h}_{{B}}{\left({t}\right)}\) is the height of the raised hands of Boris. (Express thevertical displacement in terms of H,g,and t.)
Part B.) Find the vertical displacement D(t) between the raised hands of the two players for the time period after Boris has jumped (\(\displaystyle{t}{>}{t}_{{R}}\)) but before Arabella has landed. (Express youranswer in terms of t,\(\displaystyle{t}_{{R}}\), g,and H)
Part C.) What advice would you give Arabella To minimize the chance of her shot being blocked?
Part A.) Find the vertical displacement \(\displaystyle{D}{\left({t}\right)}={h}_{{A}}{\left({t}\right)}-{h}_{{B}}{\left({t}\right)}\), as a function of time for the interval \(\displaystyle{0}{<}{t}{<}{t}_{{R}}\), where \(\displaystyle{h}_{{A}}{\left({t}\right)}\) is the height of the raised hands of Arabella, while \(\displaystyle{h}_{{B}}{\left({t}\right)}\) is the height of the raised hands of Boris. (Express thevertical displacement in terms of H,g,and t.)
Part B.) Find the vertical displacement D(t) between the raised hands of the two players for the time period after Boris has jumped (\(\displaystyle{t}{>}{t}_{{R}}\)) but before Arabella has landed. (Express youranswer in terms of t,\(\displaystyle{t}_{{R}}\), g,and H)
Part C.) What advice would you give Arabella To minimize the chance of her shot being blocked?
