Elysian Fields, Inc., uses a maximum payback period of $6$ years and currently must choose between two mutually exclusive projects. Project Hydrogen requires an initial outlay of $\$ 25,000$; project Helium requires an initial outlay of $\$ 35,000$. Using the expected cash inflows provided for each project in the accompanying table, determine each project's payback period. Which project meets Elysian's standards?

	Expected cash inflows
Year	Hydrogen	Helium
1	$\$ 6,000$	$\$ 7,000$
2	6,000	7,000
3	8,000	8,000
4	4,000	5,000
5	3,500	5,000
6	2,000	4,000
:---:	:---:	:---:
Year	Hydrogen	Helium
1	$\$ 6,000$	$\$ 7,000$
2	6,000	7,000
3	8,000	8,000
4	4,000	5,000
5	3,500	5,000
6	2,000	4,000

Form the dual problem of the linear programming problem Minimize

$$C=2 x_1+5 x_2$$

subject to

$$x_1+2 x_2 \geq 10$$

$$3 x_1+x_2 \geq 18$$

$$x_2 \geq 4$$

$$x_1, x_2 \geq 0$$

the linear programming problem has an unusual characteristic. Sketch a graph of the solution region for the problem and describe the unusual characteristic. Find the minimum and maximum values of the objective function (if possible) and where they occur. Objective function: z = x + y , Constraints: x ≥ 0 , y ≥ 0 , -x + y ≤ 1 , -x + 2y ≤ 4

One constraint to a linear programming problem is $8 x_{1}+5 x_{2}+s_{2}=59$ This constraint determines a boundary line to the feasible region. (a) Is the point (3, 7) on the boundary line? (b) Can (2, 4) be in the feasible region?

A company has a single zero coupon bond outstanding that matures in five years with a face value of $20 million. The current value of the company’s assets is$18.1 million, and the standard deviation of the return on the firm’s assets is 41 percent per year. The risk-free rate is 6 percent per year, compounded continuously.

a. What is the current market value of the company’s equity?
b. What is the current market value of the company’s debt?
c. What is the company’s continuously compounded cost of debt?
d. The company has a new project available. The project has an NPV of $2,100,000. If the company undertakes the project, what will be the new market value of equity? Assume volatility is unchanged.
e. Considering the company undertakes the new project and does not borrow any additional funds, what is the new continuously compounded cost of debt? What is happening here?

Solve the linear programming problems using graphical methods. Restrict $x \geq 0$ and $y \geq 0$.

Minimize $g=5 x+3 y$ subject to

$$\begin{array}{rr} 3 x+y \geq 12 \\ x+\ \ y \geq 6 \\ x+6 y \geq 11 \end{array}$$

Use the constraints below to solve each linear programming problem.

$$\left\{\begin{array}{rcl} x & +2y & \leq 40\\ 2 x & +y & \leq 40\\ x&+y & \geq 10 \\ & x & \geq 0 \\ & y & \geq 0 \end{array}\right.$$

Maximize $z=2x+3y$.

Use graphical methods to solve each linear programming problem.

$$\begin{array}{lr} \text{Minimize} & z=x+3 y \\ \text{subject to:} & x+y \leq 10 \\ & 5 x+2 y \geq 20 \\ & -x+2 y \geq 0 \\ & x \geq 0 \\ & y \geq 0. \end{array}$$

Find out what the model-view-controller pattern is. You can do a web search to get information, or you can use any other sources you find. How is it related to the topic discussed here? What does it suggest? How could it be applied to this project? (Only discuss its application to this project, as an actual implementation would be an advanced challenge exercise.)

The Programming Example, Converting a Number from Decimal to Binary, in Chapter 15, contains a program that uses recursion to convert a decimal number into an equivalent binary number. Write a program that uses a stack to convert a decimal number into an equivalent binary number.

A project to build a new bridge seems to be going very well since the project is well ahead of schedule and costs seem to be running very low. A major milestone has been reached where the first two activities have been totally completed and the third activity is 60 percent complete. The planners were expecting to be only 50 percent through the third activity at this time. The first activity involves prepping the site for the bridge. It was expected that this would cost $\$1,420,000$ and it was done for only $\$1,300,000$. The second activity was the pouring of concrete for the bridge. This was expected to cost $\$10,500,000$ but was actually done for $\$9,000,000$. The third and final activity is the actual construction of the bridge superstructure. This was expected to cost a total of $\$8,500,000$. To date, they have spent $5,000,000 on the superstructure. Calculate the Schedule Variance, Schedule Performance Index, and Cost Performance Index for the project to date. How is the project going?

Use graphical methods to solve each linear programming problem.

$$\begin{array}{lr} \text{Minimize} & z=4 x+7 y\\ \text{subject to:} & x-y \geq 1 \\ & 3 x+2 y \geq 18 \\ & x \geq 0 \\ & y \geq 0. \end{array}$$

Solve the linear programming problem by using the simplex method Maximize

$$\quad P=2 x_1+3 x_2$$

subject to

$$-2 x_1+x_2 \leq 2$$

$$-x_1+x_2 \leq 5$$

$$x_2 \leq 6$$

$$x_1, x_2 \geq 0$$

The most likely outcomes for a particular project are estimated as follows:

$$\begin{array}{ll} \text{Unit price:}&\$50\\ \text{Variable cost:}&\$30\\ \text{Fixed cost:}&\$300,000\\ \text{Expected sales:}&\text{30,000 units per year}\\ \end{array}$$

However, you recognize that some of these estimates are subject to error. Suppose that each variable may turn out to be either 10% higher or 10% lower than the initial estimate. The project will last for 10 years and requires an initial investment of $1 million, which will be depreciated straight-line over the project life to a final value of zero. The firm's tax rate is 35% and the required rate of return is 12%. What is project NPV in the "best-case scenario," that is, assuming all variables take on the best possible value? What about the worst-case scenario?