Consider a flat, circular current loop of radius R carrying a current I. Choose the x axis to be along the axis of the loop, with the origin at the loop’s center. Plot a graph of the ratio of the magnitude of the magnetic field at coordinate x to that at the origin for x = 0 to x = 5R. It may be helpful to use a programmable calculator or a computer to solve this problem.

Because of safety considerations, in May 2003 the Federal Aviation Administration (FAA) changed its guidelines for how small commuter airlines must estimate passenger weights. Under the old rule, airlines used 180 pounds as a typical passenger weight (including carry-on luggage) in warm months and 185 pounds as a typical weight in cold months. The Alaska Journal of Commerce (May 25, 2003) reported that Frontier Airlines conducted a study to estimate average passenger plus carry-on weights. They found an average summer weight of 183 pounds and a winter average of 190 pounds. Suppose that each of these estimates was based on a random sample of 100 passengers and that the sample standard deviations were 20 pounds for the summer weights and 23 pounds for the winter weights. a. Construct and interpret a $95\%$ confidence interval for the mean summer weight (including carry-on luggage) of Frontier Airlines passengers. b. Construct and interpret a $95\%$ confidence interval for the mean winter weight (including carry-on luggage) of Frontier Airlines passengers. c. The new FAA recommendations are 190 pounds for summer and 195 pounds for winter. Comment on these recommendations in light of the confidence interval estimates from Parts (a) and (b).

Use the Integral Test to determine whether each series converges or diverges. $\sum_{k=1}^{\infty} \frac{1}{k^{2}+1}$

Consider a bright star in our night sky. Assume its distance from the Earth is 20.0 light-years (ly) and its power output is $4.00 \times 10 ^ { 28 }$ W, about 100 times that of the Sun. Find the intensity of the starlight at the Earth.

A sample of 548 ethnically diverse students from Massachusetts were followed over a 19-month period from 1995 and 1997 in a study of the relationship between TV viewing and eating habits (Pediatrics [2003]: 1321–1326). For each additional hour of television viewed per day, the number of fruit and vegetable servings per day was found to decrease on average by 0.14 serving. a. For this study, what is the dependent variable? What is the predictor variable? b. Would the least-squares line for predicting number of servings of fruits and vegetables using number of hours spent watching TV as a predictor have a positive or negative slope? Explain.

Show that the speed of an object having momentum of magnitude p and mass m is $u = \frac { c } { \sqrt { 1 + ( m c / p ) ^ { 2 } } }$

A $\mathrm{CO}_{2}$ cartridge is used to propel a small rocket cart. Compressed gas, stored at 35 MPa and $20^{\circ} \mathrm{C}$, is expanded through a smoothly contoured converging nozzle with 0.5 mm throat diameter. The back pressure is atmospheric. Calculate the pressure at the nozzle throat. Evaluate the mass flow rate of carbon dioxide through the nozzle. Determine the thrust available to propel the cart. How much would the thrust increase if a diverging section were added to the nozzle to expand the gas to atmospheric pressure? What is the exit area? Show stagnation states, static states, and the processes on a Ts diagram. Consider the converging-diverging option. To what pressure would the compressed gas need to be raised (keeping the temperature at $20^{\circ} \mathrm{C}$) to develop a thrust of 15N? (Assume isentropic flow.)

An incompressible liquid with negligible viscosity and density $\rho=1250 \mathrm{kg} / \mathrm{m}^{3}$ flows steadily through a 5-m-long convergent-divergent section of pipe for which the area varies as $A(x)=A_{0}\left(1+e^{-x / a}-e^{-x / 2 a}\right)$ where $A_{0}=0.25 \mathrm{m}^{2}$ and a=1.5 m. Plot the area for the first 5 m. Develop an expression for and plot the pressure gradient and pressure versus position along the pipe, for the first 5 m, if the inlet centerline velocity is 10 m/s and inlet pressure is 300 kPa. Hint: Use relation $u \frac{\partial u}{\partial x}=\frac{1}{2} \frac{\partial}{\partial x}\left(u^{2}\right)$.

Consider the velocity field $\vec{V}=A\left[x^{4}-6 x^{2} y^{2}+y^{4}\right] \hat{i}+B\left[x^{3} y-x y^{3}\right] \hat{j}$; $A=2 \mathrm{m}^{-3} \cdot \mathrm{s}^{-1}$, B is a constant, and the coordinates are measured in meters. Find B for this to be an incompressible flow. Obtain the equation of the streamline through point (x, y)=(1, 2). Derive an algebraic expression for the acceleration of a fluid particle. Estimate the radius of curvature of the streamline at (x, y)=(1, 2).

Show that the nth derivative of $\left(a x^{2}+b x+c\right) e^{x}$ is a function of the same form but with different constants.

The following table gives the total world emissions of $\mathrm{CO}_{2}$ from fossil fuels, in billions of tons per year.

$$\begin{array}{c|c|c|c|c|c|c|} \hline {\text { Year }} & {1981} & {1986} & {1991} & {1996} & {2001} & {2006} & {2011} \\ \hline \mathrm{CO}_{2(\mathrm{bn}\text { tons per year) } } & {20.1} & {21.3} & {21.9} & {23.4} & {24.2} & {29.1} & {34.7} \\ \hline \end{array}$$

Use this data to estimate the total world $CO_2$ emissions between 1981 and 2011 using a right sum with n=3.

An insulation company is examining a new material for extruding into cavities. The experimental data is given below for the speed U of the upper plate, which is separated from a fixed lower plate by a 1-mm-thick sample of the material, when a given shear stress is applied. Determine the type of material. If a replacement material with a minimum yield stress of 250 Pa is needed, what viscosity will the material need to have the same behavior as the current material at a shear stress of 450 Pa?

$$\begin{matrix} \text{}{\tau(\mathrm{Pa})} & \text{50} & \text{100} & \text{150} & \text{163} & \text{171} & \text{170} & \text{202} & \text{246} & \text{349} & \text{444}\\ \text{U (m/s)} & \text{0} & \text{0} & \text{0} & \text{0.005} & \text{0.01} & \text{0.025} & \text{0.05} & \text{0.1} & \text{0.2} & \text{0.3}\\ \end{matrix}$$

For the following two-dimensional partial differential equations, find the dispersion relation: $(\mathrm{a})\ \frac{\partial^{2} u}{\partial t^{2}}=c^{2}\left(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}\right); (\mathrm{b})\ \frac{\partial^{2} u}{\partial t^{2}}=c_{1}^{2} \frac{\partial^{2} u}{\partial x^{2}}+c_{2}^{2} \frac{\partial^{2} u}{\partial y^{2}}.$

Compute the convolution of each of the following pairs of functions. $\sin a t, \sin b t, a \neq b$