**Question 1- Constraint optimization (3 marks)**

**Solve this problem with the aid of http://www.wolframalpha.com/**

A company allocates $600,000 to spend on advertising and research. The company estimates that by spending thousand dollars on advertising and thousand dollars on research, they will sell a total of approximately units of its product. How much should the company spend on research and advertising in order to maximize sales?

Note:

**In your report include:**

- Formal presentation of the optimisation problem
- Step by stem implementation process in
**http://www.wolframalpha.com/** - The screenshot of your input page
- The screenshot of your output page
- 300 words report of your results, including the sensitivity analysis using the Lagrange multiplier

**Question 2- (1.5 marks)**

In economic theory, Price equilibrium is found where supply and demand are equal. Suppose, the demand and supply equations for a good are given by and . Determine the equilibrium price and quantity.

**In your report include:**

- Use matrix algebra for your answer

**Question 3- Algebra (2 marks)**

The percentage, y, of Europeans possessing a mobile phone years after it was introduced is modelled by

**In your report include:**

- Find the percentage of Europeans that have mobile phones
- at the launch of the product;
- after 3 years;

- Use Excel spread sheet to Sketch the graph of y against t. (Do not include the spread sheet in the report. Just import the graph in the report)
- After how many years will the percentage of Europeans possessing mobile phones first reach 75%?

**Question 4- Loan Amortization (3.5 marks)**

**Preliminary: **To be able to complete this question, you need to research what is loan an amortization schedule and how it works. You are not required to include this part in your report.

**Requirement: **Use an Excel Spreadsheet to create a loan repayment schedule calculator. Your calculator must take the following input and generate the following output.

**Input:**

- Loan amount
- Interest rate p.a. (fixed rate, not variable rate)
- Payment periods (monthly, fortnightly or weekly)
- Term of the loan in years (max 30 years)

**Output and other requirement:**

- The periodic payments
- The decomposition of principle and interest payment per period
- Based on the term of the loan and the payment periods, your spread sheet must adjust the number of required fields. Example: a 20 years fortnightly loan will require rows, while a 25 years monthly loan will require
- Total interest paid for the whole loan

Additionally, your calculator must allow for irregular additional payments. That would impact your principle payment and the term of your loan. In this case, your calculator must show how much interest you will be saving, if you make additional payments.