The simplest cost function is the linear cost function, $C(x) = mx + b$, where the y-intercept b represents the fixed costs of operating a business and the slope m represents the cost of each item produced. Suppose that a small bicycle manufacturer has daily fixed costs of $1800 and each bicycle costs$90 to manufacture. Graph the model.

After a great deal of experimentation, two Atlantic Institute of Technology senior physics majors determined that when a bottle of French champagne is shaken several times, held upright, and uncorked, its cork travels according to

$$s(t)=-16 t^2+40 t+3,$$

where s is its height (in feet) above the ground t seconds after being released. (a) How high will it go? (b) How long is it in the air?

Complete the problem.

$$\begin{array}{|c| c c c c c c c |c|}\hline \small \textbf{Total} &\small \textbf{(Selling Price per Share} & \times & \small\textbf{Number of Shares} & - & \small\textbf{Commission)}& =& \small\textbf{Net Sale} &\small \textbf{Amount of}\\ \small\textbf{Paid} & & & & & & & &\small\textbf{Profit or Loss}\\ \hline \$3,250 & (\$18 & \times & 200) & - & \$44.90 &=& ? & ?\\ \hline \end{array}$$

Chinese online retailing giant Alibaba recorded over $\$ 240$ billion in sales in $2013$ $-$ more than Amazon and eBay combined. On its busiest day of the year, Alibaba sold $\$ 5.75$ billion of goods, three times more than what all of America spent on all US shopping sites combined in the same $24$ hours. Alibaba is planning to become a public limited company by selling shares on the New York Stock Exchange. This will value the company at $\$ 150$ billion or more. With the capital it raises, Alibaba has plans not only to dominate China, but to launch its own US marketplace.

At first Alibaba's business was a B$2$B marketplace connecting American merchants with Chinese suppliers. It quickly expanded into retail B$2$C operations with Taoboa, where eight million small businesses now list their items. It later up-scaled with Tmall, a shopping site where premium brands such as Apple, Nike and Disney host virtual shops for Chinese consumers. During the last nine months of $2013$ Alibaba made over $\$ 6.5$ billion dollars in revenue from selling fees, nearly half of which was profit. Alibaba's greatest challenge in entering the US market will be to gain customers trust. In the past it's been criticised for allowing dishonest businesses to sell illegal counterfeit goods on its sites. Customers searching for products have to go directly to Alibaba's sites as, unlike most e-commerce companies, it doesn't permit search engines to index its products. Merchants are able list for free on sites like Taobao, but it charges them a premium if they want their goods promoted in its search engine.

One New York customer has used Alibaba on several occasions in recent years. He said that although he wasn't sure whether to trust the site at first, in the end the prices for goods were too competitive to ignore. He was able to buy the products direct from the manufacturers at $10 \%$ of the price he would have paid on a US website.

The arrival of Alibaba will be a worrying prospect for Amazon, which uses its ability to undercut its competition to continue to build its business.

Outline one problem businesses might have using Alibaba as a way of buying and selling goods.

Complete the problem.

$$\begin{array}{|c| c c c c c c c |c|}\hline \small \textbf{Total} &\small \textbf{(Selling Price per Share} & \times & \small\textbf{Number of Shares} & - & \small\textbf{Commission)}& =& \small\textbf{Net Sale} &\small \textbf{Amount of}\\ \small\textbf{Paid} & & & & & & & &\small\textbf{Profit or Loss}\\ \hline \$5925 & (\small \$52 \frac{1}{4} & \times & 100) & - & \$59.60 &=& ? & ?\\ \hline \end{array}$$

In physics it is shown that the height s of a ball thrown straight up with an initial speed of 96 ft sec from a rooftop 112 feet high is $s=s(t)=-16 t^{2}+96 t+112$ where t is the elapsed time that the ball is in the air. The ball misses the rooftop on its way down and eventually strikes the ground. What is the average speed of the ball from t = 0 to t = 2?

Hansus Enterprises is a start-up small business specializing in software systems for machine based high-school student tutoring in science and math. Sheryl Hansus, the owner, has spoken with lenders to obtain a loan of $50,000 now (year 0), and the same amount in years 3 and 6. Two different repayment schedules are available. They are: Schedule A: Pay a uniform amount of$19,500 in years 3 through 12. Schedule B: Pay a uniform amount of $20,000 in years 1 through 6, with “balloon” payments of an additional$20,000 in year 2 and a final amount of $40,000 in year 7. To assist in the decision of which schedule to select, determine the following: (a) Total amount repaid for each schedule. Which is smaller? (b) The equivalent annual worth of the loan amounts and of the repayments at i = 5% per year over a 12-year evaluation period. Which schedule has the smaller loss per year? Why is this the case?

Which of the following is most likely to happen under hypnosis? a. A person, will remember a traumatic event from their past that they had forgotten. b. An adult, will regress to acting exactly like they did as a child. c. A person may be unable to report their grandmother's name if asked to forget it. d. An eyewitness to a crime will give a vivid and accurate description of events that they had forgotten. e. A person, will become less open to suggestions.

Add.

$$\small \frac{3}{4} +\small \frac{1}{8}$$

The hyperbolic cosine of x, denoted by $\cosh x$, is defined by

$$\cosh x=\frac{1}{2}\left(e^r+e^{-r}\right) \text {. }$$

This function occurs often in physics and probability theory. The graph of $y=\cosh x$ is called a catenary.

(a) Use differentiation and the definition of a Taylor series to compute the first four nonzero terms in the Taylor series of $\cosh x$ at x=0.

(b) Use the known Taylor series for $e^x$ to obtain the Taylor series for $\cosh x$ at x=0.

You start a small beekeeping business. Your initial costs are $500 for equipment and bees. You estimate it will cost$1.25 per pound to collect, clean, bottle, and label the honey. How many pounds of honey must you product before your average cost per pound is $1.79? Justify your answer.

The simplest cost function is the linear cost function, $C(x) = mx + b$, where the y-intercept b represents the fixed costs of operating a business and the slope m represents the cost of each item produced. Suppose that a small bicycle manufacturer has daily fixed costs of $1800 and each bicycle costs$90 to manufacture. How many bicycles could be manufactured for $3780?

Tom’s Tax Services is a small accounting firm that offers tax services to small businesses and individuals. A local store owner has approached Tom about doing his taxes but is concerned about the fees Tom normally charges. The costs and revenues at Tom’s Tax Services are presented on the following page.

If Tom gets the store’s business, he will incur an additional $60,000 in labor costs. Tom also estimates that he will have to increase equipment leases by about 10 percent, supplies by 5 percent, and other costs by 15 percent.

Required
What are the differential costs that would be incurred as a result of adding this new client?