Doctors Hanson, Dominick, and Borchard are radiologists living in Fargo, North Dakota. They realize that many of the state's small, rural hospitals cannot afford to purchase their own magnetic resonance imaging devices (MRIs). Thus, the doctors are considering whether it would be feasible for them to form a corporation and invest in their own mobile MRI unit. The unit would be transported on a scheduled basis to more than 100 rural hospitals using an 18-wheel tractor-trailer. The cost of a tractor-trailer equipped with MRI equipment is approximately $\$ 1,250,000$. The estimated life of the investment is eight years, after which time its salvage value is expected to be no more than $\$ 100,000$. The doctors anticipate that the investment will generate incremental revenue of $\$ 800,000$ per year. Incremental expenses (which include depreciation, insurance, fuel, maintenance, their salaries, and income taxes) will average $\$ 700,000$ per year. Net incremental cash flows will be reinvested back into the corporation. The only difference between incremental cash flows and incremental income is attributable to depreciation expense. The doctors require a minimum return on their investment of 12 percent.

Compute the payback period of the mobile MRI proposal.

For elliptic curves, there are nice ways of finding points with rational coordinates (see Ezra Brown’s article “Three Fermat Trails to Elliptic Curves” in the May 2000 College Mathematics Journal). Show that the points (−3, 0) and (0, 3) are on the elliptic curve defined by $y^{2}=x^{3}-6 x+9$. Find the line through these two points and showthat the line intersects the curve in another point with rational (in this case, integer) coordinates.

Four hundred people were selected from each of the four geographic regions (Midwest, Northeast, South. West) of the United States, and they were asked which form of camping they prefer. Pop-up camper/trailer, family-style (tenting with restrooms), rustic (tenting, no restrooms), or none were the options. The following table displays the survey's findings.

$$\begin{array}{lcccr} \hline & \text { Midwest } & \text { Northeast } & \text { South } & \text { West } \\ \hline \text { Camper/trailer } & 132 & 129 & 129 & 135 \\ \text { Family style } & 180 & 175 & 168 & 146 \\ \text { Rustic } & 46 & 50 & 59 & 68 \\ \text { None } & 42 & 46 & 44 & 51 \\ \hline \end{array}$$

Can you conclude that the distributions of preferred camping types are different for at least two of the regions based on the evidence from these samples? Use $\alpha=.01$.

From the trailer hitch from the figure has loads applied as shown in the figure. The tongue weight of $980 \mathrm{~N}$ acts downward and the pull force of $4905 \mathrm{~N}$ acts horizontally. Using the dimensions of the ball bracket in the figure, draw a free-body diagram of the ball bracket and find the tensile and shear loads applied to the two bolts that attach the bracket to the channel in the figure, determine the horizontal force that will result on the ball from an impact between the ball and the tongue of the $2000-\mathrm{kg}$ trailer if the hitch deflects $2.8 \mathrm{~mm}$ dynamically on impact. The tractor weighs $9807 \mathrm{~N}$. The velocity at impact is $0.3 \mathrm{m/sec}$. (also see the figures) determine the contact stresses in the ball and the ball cup (not shown). Assume that the ball is $2-\mathrm{in} \mathrm{~dia}$ and the ill-fitting ball cup that surrounds it has an internal spherical surface $10\%$ larger in diameter than the ball.

Doctors Hanson, Dominick, and Borchard are radiologists living in Fargo, North Dakota. They realize that many of the state's small, rural hospitals cannot afford to purchase their own magnetic resonance imaging devices (MRIs). Thus, the doctors are considering whether it would be feasible for them to form a corporation and invest in their own mobile MRI unit. The unit would be transported on a scheduled basis to more than 100 rural hospitals using an 18-wheel tractor-trailer. The cost of a tractor-trailer equipped with MRI equipment is approximately $\$ 1,250,000$. The estimated life of the investment is eight years, after which time its salvage value is expected to be no more than $\$ 100,000$. The doctors anticipate that the investment will generate incremental revenue of $\$ 800,000$ per year. Incremental expenses (which include depreciation, insurance, fuel, maintenance, their salaries, and income taxes) will average $\$ 700,000$ per year. Net incremental cash flows will be reinvested back into the corporation. The only difference between incremental cash flows and incremental income is attributable to depreciation expense. The doctors require a minimum return on their investment of 12 percent.

Compute the return on average investment of the proposal.

A food company manufactures five types of 8 oz Trail mix packages using different mixtures of peanuts, almonds, walnuts, raisins, and M&Ms. The mixtures have the following compositions: $$ \begin{matrix} \text{ } & \text{Peanuts (oz)} & \text{Almonds (oz)} & \text{Walnuts (oz)} & \text{Raisins (oz)} & \text{M&Ms(oz)}\\ \text{Mix 1} & \text{3} & \text{1} & \text{1} & \text{2} & \text{1}\\ \text{Mix 2} & \text{1} & \text{2} & \text{1} & \text{3} & \text{1}\\ \text{Mix 3} & \text{1} & \text{1} & \text{0} & \text{3} & \text{3}\\ \text{Mix 4} & \text{2} & \text{0} & \text{3} & \text{1} & \text{2}\\ \text{Mix 5} & \text{1} & \text{2} & \text{3} & \text{0} & \text{2}\\ \end{matrix} $$ How many packages of each mix can be manufactured if 128 lb of peanuts, 118 lb of almonds, 112 lb of walnuts, 112 lb of raisins, and 104 lb of M&Ms are available? Write a system of linear equations and solve.

Suppose that you are driving an 18-wheeler tractor-trailer truck along the highway. A car pulls up alongside your truck and then moves ahead very slowly. As it passes, the combined sound of the car and truck engines pulsates louder and softer. The combined sound can be either a sum of the two sinusoids or a product of them.

At what rate is the car's engine rotating?

A model airplane descends in a vertical dive through air at standard conditions ( 1 atmosphere and $20^{\circ} \mathrm{C}$ ). The majority of the drag is due to skin friction on the wing (like that on a flat plate). The wing has a span of $1 \mathrm{~m}$ (tip to tip) and a chord length (leading edge to trailing edge distance) of $10 \mathrm{~cm}$. The leading edge is rough, so the turbulent boundary layer is "tripped." The model weighs $3 \mathrm{~N}$. Find the speed (in meters per second) at which the model will fall.

Fission tracks are trails found in uranium-bearing minerals, left by fragments released during fission events. The article “Yo-yo Tectonics of the Nigde Massif During Wrenching in Central Anatolia” reports that fifteen tracks on one rock specimen had an average track length of $13\mu m$ with a standard deviation of $2 \mu \mathrm{m}.$ Assuming this to be a random sample from an approximately normal population, find a 99% confidence interval for the mean track length for this rock specimen.

An airplane wing of 2 m chord length (leading edge to trailing edge distance) and 11 m span flies at 200 km/hr in air at 30$^{\circ}C$. Assume that the resistance of the wing surfaces is like that of a flat plate. a. What is the friction drag on the wing? b. What power is required to overcome this? c. How much of the chord is laminar? d. What will be the change in drag if a turbulent boundary layer is tripped at the leading edge?

When Earth traveled through the storm of meteorites trailing the comet Tempel-Tuttle on November 17, 1998, the storm was 1,000 times as intense as the average meteor storm. Before the comet arrived, telecommunication companies worried about the potential damage that might be inflicted on the approximately 650 satellites in orbit. It was estimated that each satellite had a 1% chance of being hit, causing damage to the satellite’s electronic system. One company had five satellites in orbit at the time. Determine the probability distribution of the number of the company’s satellites that would be damaged.

A model airplane has a wing span of $6\ \mathrm{ft}$ and a chord (leading edge-trailing edge distance) of $4.5 \mathrm{in}$. The model flies in air at $60^{\circ} \mathrm{F}$ and atmospheric pressure. The wing can be regarded as a flat plate so far as drag is concerned. (a) At what speed will a turbulent boundary layer start to develop on the wing? (b) What will be the total drag force on the wing just before turbulence appears?

A tractor-trailer truck is coasting freely, with no brakes, down an $8^{\circ}$ slope at $1000$-$\mathrm{m}$ standard altitude. Rolling resistance is $120 \mathrm{~N}$ for every $\mathrm{m} / \mathrm{s}$ of speed. Its frontal area is $9 \mathrm{~m}^2$, and the weight is $65\ \mathrm{kN}$. Compute the terminal coasting velocity, in $\mathrm{mi} / \mathrm{h}$, for $(a)$ no deflector and (b) a deflector installed.

Here are three situations in which consumers might need some protection. A motorist asks for a tow rope 'strong enough for my trailer'. It breaks the first time he tries to use it. Do you think the consumer needs some legal protection in these cases? For each example. identify which part of UK consumer protection laws have broken by the business selling the goods.