Question

1, You own a hamburger franchise and are planning to shut down operations for the day, but you are left with 11 buns, 13 defrosted beef patties, and 10 opened cheese slices. Rather than throw them out, you decide to use them to make burgers that you will sell at a discount. Plain burgers each require 1 beef patty and 1 bun, double cheeseburgers each require 2 beef patties, 1 bun, and 2 slices of cheese, while regular cheeseburgers each require 1 beef patty, 1 bun, and 1 slice of cheese. How many of each should you make? HINT [See Example 1.]

plain burgersdouble cheeseburgersregular cheeseburgers

2, Inverse mutual funds, sometimes referred to as "bear market" or "short" funds, seek to deliver the opposite of the performance of the index or category they track, and can thus be used by traders to bet against the stock market. This question is based on the following table, which shows the performance of three such funds as of February 27, 2015.

	Year-to-date Loss
SHPIX (Short Smallcap Profund)	4%
RYURX (Rydex Inverse S&P 500)	3%
RYCWX (Rydex Inverse Dow)	6%

You invested a total of $11,000 in the three funds at the beginning of 2015, including an equal amount in RYURX and RYCWX. Your year-to-date loss from the first two funds amounted to $290. How much did you invest in each of the three funds?

SHPIX$

RYURX$

RYCWX$

At the end of a certain year, the four companies with the largest number of home Internet users in the United States were Microsoft, Time Warner, Yahoo, and Google, with a combined audience of 284 million users. Taking x to be the Microsoft audience in millions, y the Time Warner audience in millions, z the Yahoo audience in millions, and u the Google audience in millions, it was observed that

					z	−	u	=	3(x − y) + 22
					x	+	y	=	44 + z + u
and	x	−	y	+	z	−	u	=	46.

3, How large was the audience of each of the four companies at the end of that year?

Microsoft	million
Time Warner	million
Yahoo	million
Google	million

4, In the 1990s, significant numbers of tourists traveled from North America and Asia to Australia and South Africa. In 1998, a total of 2,231,000 of these tourists visited Australia, while 388,000 of them visited South Africa. Also, 630,000 of these tourists came from North America, and a total of 2,619,000 tourists traveled from these two regions to these two destinations. (Assume no single tourist visited both destinations or traveled from both North America and Asia.)

(a) The given information is not sufficient to determine the number of tourists from each region to each destination. Why?

This system has no solution.This system has infinitely many solutions.    This system has finite number of solutions.This system has one solution.

(b) If you were given the additional information that a total of 1,989,000 tourists came from Asia, would you now be able to determine the number of tourists from each region to each destination?

YesNo    

If so, what are these numbers? (If the given information is not sufficient, enter NONE in all answer blanks.)

from North America to Australia	tourists
from North America to South Africa	tourists
from Asia to Australia	tourists
from Asia to South Africa	tourists

(c) If you were given the additional information that 199,000 tourists visited South Africa from Asia, would you now be able to determine the number of tourists from each region to each destination?

YesNo    

If so, what are these numbers? (If the given information is not sufficient, enter NONE in all answer blanks.)

from North America to Australia	tourists
from North America to South Africa	tourists
from Asia to Australia	tourists
from Asia to South Africa	tourists

Answer 1

1.

Let,

x1 = number of plain burgers to make
x2 = number of double cheeseburgers to make
x3 = number of white regular cheeseburgers to make

objective is to produce as much as possible = Max x1+x2+x\x3

subject to,

x1+2x2+x3 <= 13 (Beef patties)
x1+x2+x3 <= 11 (Buns)
2x2+x3 <= 10 (Cheese slice)

x1,x2,x3 >= 0 (non-negativity constraint)

Solving in solver we get,

x1 = number of plain burgers to make = 11
x2 = number of double cheeseburgers to make = 0
x3 = number of white regular cheeseburgers to make = 0

Solver screenshot

Solver formula