For the given problem, consider the homogeneous systems of linear equations. For convenience, general solutions are given. It can be shown, as in the previous exercise, that the sets of solutions form subspaces of $\mathrm{R}^4$.

(a) Use the general solution to construct two specific solutions. (b) Use the operations of addition and scalar multiplication to generate four vectors from these two solutions. (c) Use the general solution to check that these vectors are indeed solutions, giving the values of $r$ and $s$ for which they are solutions.

$$\begin{aligned} x_1+x_2+x_3+3 x_4&=0\\ x_1+2 x_2+4 x_3+5 x_4&=0\\ x_1 \quad-2 x_3+x_4&=0 \end{aligned}$$

General solution is $(2 r-s,-3 r-2 s, r, s)$.

(a) Evaluate the equation: $h(x)=\left(\sqrt{x^4+1}-1\right) / x^4$ for $x=1,0.5$, $0.2$, and $0.1$. (b) Guess the value of $\lim _{x \rightarrow 0} \frac{\sqrt{x^4+1}-1}{x^4}$. (c) Evaluate $h(x)$ for $x=0.05,0.01,0.001$, and $0.0001$. Are you still confident that your guess in part (b) is correct? (d) Explain why you eventually obtained 0 values in part (c). It may help to evaluate just the numerator of $h(x)$ for $x=0.001$ and $0.0001$. Is your calculator giving you the correct values? (e) Graph the function $h$ in the viewing rectangle $[-1,1]$ by $[0,1]$. Does the graph corroborate your guess from part (b)? Then change the viewing rectangle to $[-0.005,0.005]$ by $[0.48,0.52]$. Can you explain the appearance of the graph?

The demand for nurses between 2000 and 2015 is estimated to be $D(t)=0.0007 t^{2}+0.0265 t+2$ $(0 \leq t \leq 15)$ where D(t) is measured in million and t=0 corresponds to the year 2000. The supply of nurses over the same time period is estimated to be $S(t)=-0.0014 t^{2}+0.0326 t+1.9$ $(0 \leq t \leq 15)$ where S(t) is also measured in millions. a. Find an expression G(t) giving the gap between the demand and supply of nurses over the period in question. b. Find the interval where G is decreasing and where it is increasing. Interpret your result. c. Find the relative extreme of G. Interpret your result.

Angela Corporation issues 2,000 convertible bonds at January 1, 2016. The bonds have a 3-year life, and are issued at par with a face value of $1,000 per bond, giving total proceeds of$2,000,000. Interest is payable annually at 6%. Each bond is convertible into 250 ordinary shares (par value of $1). When the bonds are issued, the market rate of interest for similar debtwithout the conversion option is 8%.

Instructions

• a. Compute the liability and equity component of the convertible bond on January 1, 2016.
• b. Prepare the journal entry to record the issuance of the convertible bond on January 1, 2016.
• c. Prepare the journal entry to record the repurchase of the convertible bond for cash at January 1, 2019, its maturity date.

China's push to install more wind energy capacity has started paying off. As of 2012, wind energy had become the country's third largest source of energy, after coal and hydroelectric power. The amount of wind energy generated in China from 2005 through 2012 grew at the rate of $r(t)=5.018 t-3.204 \ (0 \leq t \leq 7)$ terawatt-hours/year, in year t, where t = 0 corresponds to 2005. The amount of wind energy generated in 2005 stood at 1.8 terawatt-hours. a. Find an expression giving the amount of wind energy generated in year t. b. How much wind energy was generated in 2012? c. If the trend continued into 2013, how much wind energy was generated in China in that year?

During the nineteenth century, the U.S. Congress encouraged railroad companies to build transcontinental railways across the Great Plains by giving them land grants. At that time, the federal government owned most of the land on the Great Plains. The land grants consisted of the land on which the railway was built and alternating sections of 1 square mile each on either side of the railway to a distance of 6 to 40 miles, depending on the location. The railroad companies were free to sell this land to farmers or anyone else who wanted to buy it. The process of selling the land took decades. Some economic historians have argued that the railroad companies charged lower prices to ship freight because they owned so much land along the tracks. Briefly explain the reasoning of these economic historians.

Medway Printers (MP) manufactures printers. Assume that MP recently paid $900,000 for a patent on a new laser printer. Although it gives legal protection for 20 years, the patent is expected to provide a competitive advantage for only eight years. Requirements

1. Assuming the straight-line method of amortization, make journal entries to record (a) the purchase of the patent and (b) amortization for the first full year.
2. After using the patent for four years, M P learns at an industry trade show that another company is designing a more efficient printer. On the basis of this new information, MP decides, starting with year 5, to amortize the remaining cost of the patent over two remaining years, giving the patent a total useful life of six years. Record amortization for year 5.

A friend of mine is giving a dinner party. His current wine supply includes 8 bottles of zinfandel, 10 of merlot, and 12 of cabernet (he only drinks red wine), all from different wineries. a. If he wants to serve 3 bottles of zinfandel and serving order is important, how many ways are there to do this? b. If 6 bottles of wine are to be randomly selected from the 30 for serving, how many ways are there to do this? c. If 6 bottles are randomly selected, how many ways are there to obtain two bottles of each variety? d. If 6 bottles are randomly selected, what is the probability that this results in two bottles of each variety being chosen? e. If 6 bottles are randomly selected, what is the probability that all of them are the same variety?

Solve the following equations in the given interval, giving your answers to 3 significant figures. (a) (i)

$$\cos 2 x = \frac { 1 } { 3 } \text { for } 0 ^ { \circ } \leq x \leq 360 ^ { \circ }$$

(ii)

$$\cos 3 x = \frac { 2 } { 5 } \text { for } 0 ^ { \circ } \leq x \leq 360 ^ { \circ }$$

(b) (i)

$$\sin ( 3 x - 1 ) = - 0.2 \text { for } 0 \leq x \leq \pi$$

(ii)

$$\sin ( 2 x + 1 ) = \frac { 2 } { 3 } \text { for } 0 \leq x \leq 2 \pi$$

(c) (i)

$$\tan \left( x - 45 ^ { \circ } \right) = 2 \text { for } - 180 ^ { \circ } \leq x \leq 180 ^ { \circ }$$

(ii)

$$\tan \left( x + 60 ^ { \circ } \right) = - 3 \text { for } - 180 ^ { \circ } \leq x \leq 180 ^ { \circ }$$

Select the best answer. A researcher claims to have found a drug that causes people to grow taller. The coach of the basketball team at Brandon University has expressed interest but demands evidence. Over 1000 Brandon students volunteer to participate in an experiment to test this new drug. Fifty of the volunteers are randomly selected, their heights are measured, and they are given the drug. Two weeks later, their heights are measured again. The power of the test to detect an average increase in height of 1 inch could be increased by (a) using only volunteers from the basketball team in the experiment. (b) using

$$\alpha = 0.01$$

instead of

$$\alpha = 0.05.$$

(c) using

$$\alpha = 0.05$$

instead of

$$\alpha = 0.01.$$

(d) giving the drug to 25 randomly selected students instead of 50. (e) using a two-sided test instead of a one-sided test.

The elasticity and bulk modulus of air can be measured using an apparatus consisting of a vertical glass tube and a ball bearing that just fits in the inner diameter of the tube. A $110$-g ball bearing is dropped into the top of a $47-\mathrm{cm}$-high glass tube of diameter $3.0 \mathrm{~cm}$. The ball is observed to oscillate as if the air column were a spring, $59$ times in $10$s giving a period of $0.17\mathrm{s}$.

(a) Find the effective spring constant of the air column.

(b) Show that this effective spring constant is related to the bulk modulus by $k=A^2 B / V$, where $A$ is the cross-sectional area of the tube and $V$ is the volume of the tube. Hint: Start with Equation $11.25$ and express the change in volume in terms of the displacement of the mass.

(c) Determine the bulk modulus of air.

Sikhism, a religion founded in the 15 th century in India, is going through turmoil due to a rapid decline in the number of Sikh youths who wear turbans (The Washington Post, March 29,2009 ). The tedious task of combing and tying up long hair and a desire to assimilate has led to approximately $25 \%$ of Sikh youths giving up the turban.

What is the probability that exactly two in a random sample of five Sikh youths wear a

a. turban?

b. What is the probability that two or more in a random sample of five Sikh youths wear a turban?

c. What is the probability that more than the expected number of Sikh youths wear a turban in a random sample of five Sikh youths?

d. What is the probability that more than the expected number of Sikh youths wear a turban in a random sample of 10 Sikh youths?

Apply the grid method to each situation below.

a. Use the mentioned Eq. to predict the lift force in newtons for a spinning baseball. Use a coefficient of lift of $C_L=1.2$. The speed of the baseball is $90 \mathrm{mph}$. Calculate area using $A=\pi r^2$, where the radius of a baseball is $r=1.45$ in. Assume a hot summer day.

b. Use the mentioned Eq. to predict the size of a wing in $\mathrm{mm}^2$ needed for a model aircraft that has a mass of $570 \mathrm{~g}$. Wing size is specified by giving the wing area $(A)$ as viewed by an observer looking down on the wing. Assume the airplane is traveling at $80 \mathrm{mph}$ on a hot summer day. Use a coefficient of lift of $C_L=1.2$. Assume straight and level flight so lift force balances weight.

A scientist collected the following data on the speed, in centimeters per second, at which ants ran at the given ambient temperature, in degrees Celsius. ${ }^{23}$ a. Discover the equation of the regression line, giving the speed as a function of the temperature.

$$\begin{array}{|c|c|} \hline \text { Temperature } & \text { Speed } \\ \hline 25.6 & 2.62 \\ \hline 27.5 & 3.03 \\ \hline 30.3 & 3.57 \\ \hline 30.4 & 3.56 \\ \hline 32.2 & 4.03 \\ \hline 33.0 & 4.17 \\ \hline 33.8 & 4.32 \\ \hline \end{array}$$

b. Describe in practical terms the meaning of the slope of the regression line. c. Express, using functional notation, the speed at which the ants run when the ambient temperature is 29 degrees Celsius, and then evaluate that value. d. The scientist observes the ants running at a speed of $2.5$ centimeters per second. Find the ambient temperature.