Iterations Business#

Section Title: Iterations Business

Business Applications#

Question-1: Max of Revenue#

Revenue (R) is the product of the number (n) of items sold and the price (p) of the item.

  • \(R = p\cdot n\)

The number of item sold is given by the following equation: \(n = -1.5p+30\)

Write a program which finds the \(p\) which maximize the revenue for \(0\le p \le 20\)

  • Plot the graph of revenue for \(0\le p \le 20\)

Question-2: Profit#

Profit (P) is the difference between Revenue and Cost

  • \(Profit = Revenue - Cost\)

The monthly fixed cost of a cable factory is 1500 dollars. Each cable costs 13 dollars and sells for 25 dollars.

  • Find the profit of producing 400 cables.

  • For what number of cables produced profit is zero.

    • Hint: Calculate the profit for \(n\) between 1 and 400.

Question-3: Linear Depreciation#

The value of a new machine is \(100,000\) dollars and its values is depreciated by \(7500\) dollars per year. After how many years the value of the machine will be 2,500 dollars.

  • Hint: Calculate the value for years between 1 and 100.

Question-4: Equilibrium Point#

An equilibrium point is where the demand and supply curves intersect.

For the given demand and the supply functions find the equilibrium point.

Demand: \(p(q) = -0.03q + 1000\)

Supply: \(p(q) = 0.02q + 400\)

  • Hint: Calculate the demand and supply for \(q\) between 1 and 20,000.

Question-5: Linear Programming#

Linear programming involves finding the highest or lowest possible outcome of a linear function, while satisfying specific constraints.

The store sells three items labeled (a, b, c) priced at 2, 3, and 4 dollars respectively. The following conditions are known:

  • The store can sell up to 100 units of each item.

  • The combined sales of items b and c exceed 90 units.

  • The total sales of items a and b do not surpass 50 units.

  • The sales of items a and c are more than 80 units.

What is the highest revenue achievable, and at what quantities of items is this maximum revenue attained?

Question-6: Home Mortgage#

Amortization is paying off a loan by making equal payments.

The present value of an amortized loan is given by the following formula: \(\displaystyle P = R \left( \frac{1-(1+\frac{r}{m})^{-mt}}{\frac{r}{m}} \right)\) where,

  • \(P\): Present value

  • \(R\): regular payment

  • \(r\): annual interest rate

  • \(m\): number of payments in a year

  • \(t\): number of years

If you take out an 449,000 dollars loan to buy a house with an annual interest rate of 2.75%, to be paid off over 30 years, what is the monthly payment?

  • Sketch the graph of monthly payments for annual interest rates of 2%, 3%, 4%, 5%, 6%, 7%, 8%, 9%, and 10%.

Hint:

  • Use the variables \(P, r, m, t\) for the given values.

  • Calculate the value of the variable \(R\).

Question-7: Amortization Schedule-1#

If you take out a 449,000 dollars loan to buy a house with an annual interest rate of 2.75%, to be paid off over 30 years with a monthly payment of $1,833, construct an amortization schedule for the first 5 payments.

For each month, a fixed monthly payment is deducted from the balance. Additionally, simple interest is added to the loan based on that month’s remaining balance.

Hint:

  • Use the variables \(P, r, m, t, R\) for the given values.

  • Calculate the monthly simple interest for each monthly balance.

Question-8: Amortization Schedule-2#

If you take out a 449,000 dollars loan to buy a house with an annual interest rate of 2.75%, to be paid off over 30 years with a monthly payment of 1,833 dollars, construct an amortization schedule for the last 5 payments.

For each month, a fixed monthly payment is deducted from the balance. Additionally, simple interest is added to the loan based on that month’s remaining balance.