Variables Business

Business Applications

Question-1: Demand Function

The demand, p = D(q), function shows how much of a product consumers want (q=quantity) and the price (p) they are willing to pay.

The demand function is given as the following linear relation: \(p = -2q+300\)

• Determine the price if the number of items demanded is 75.

• Use the variables \(q\) and \(p\).

Question-2: Demand Equation

A store can sell 1200 ice creams at 3 dollars each. If the price is 5 dollars, they can sell 800 ice creams.
What will be the price if 500 ice creams are sold, assuming the demand is linear?

Hint:

• Since the relation between p and q is linear, we have \(p=mq+b\)

• The slope \(m\) is calculated using the formula \(\displaystyle m=\frac{p_2-p_1}{q_2-q_1}\) for given two points \((q_1,p_1)\) and \((q_2,p_2)\).

• The y-intercept \(b\) can be found using one of the points \((q_1,p_1)\) or \((q_2,p_2)\) so \(b=p_1 - mq_1\) or \(b=p_2 - mq_2\)

• Use the variables \(q_1, p_1, q_2, p_2\) for the given values in the question.

Question-3: Supply Function

The supply function, \(p=S(q)\), shows how the price (p) affects the quantity (q) that producers want to supply.

The supply function is given by the linear relation \(p = 4q+100\).

• Determine the price when 120 items are supplied.

• Use the variables \(q\) and \(p\).

Question-4: Supply Equation

A store can supply 2000 ice creams at 4 dollars each. If the price is 6 dollars, they can supply 6000 ice creams. How many ice creams can they supply at 5 dollars, assuming the supply is linear?

Hint:

• Since the relation between \(p\) and \(q\) is linear, we have \(p=mq+b\)

• The slope \(m\) is calculated using the formula \(\displaystyle m=\frac{p_2-p_1}{q_2-q_1}\) for given two points \((q_1,p_1)\) and \((q_2,p_2)\).

• The y-intercept \(b\) can be found using one of the points \((q_1,p_1)\) or \((q_2,p_2)\) so \(b=p_1 - mq_1\) or \(b=p_2 - mq_2\)

• Use the variables \(q_1, p_1, q_2, p_2\) for the given values in the question.

Question-5: Simple Interest

Simple Interest Formulas: \(\displaystyle I=P r t\) and \(A = P + I\) where

• A is the final amount

• I is the interest amount

• P is the initial (principal) amount

• r is the annual interest rate (decimal form)

• t is the number of years

You borrow 1,200 dollars for 8 months at a 24% annual simple interest rate.

• How much interest will you pay?

• What is the total amount you will repay after 8 months?

Hint:

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

• Calculate the values of the variables \(I\) and \(A\).

• t is 8/12 years.

Output:
a) I = 192.0
b) A = 1392.0

Question-6: Compound Interest

Compund interest formula: \(\displaystyle A=P\left(1+{\frac {r}{n}}\right)^{nt}\) where

• A is the final amount

• P is the initial (principal) amount

• r is the annual interest rate (decimal form)

• n is the compounding frequency (anual: n=1, semi annual: r=2, quarter: r=4, monthly: r=12)

• t is the number of years

If you deposit 800 dollars into a savings account with a 2.75 percent interest rate compounded monthly, what will be your balance after five years (Round your answer to the nearest hundredth)?

Hint:

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

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

Question-7: Continuous Compound Interest

Continuous Compound Interest formula: \(\displaystyle A=Pe^{rt}\) where

• A is the final amount

• P is the initial (principal) amount

• r is the annual interest rate (decimal form)

• t is the number of years

If you deposit 500 dollars into a savings account with a 5.25 percent interest rate compounded continuously, what will be your balance after ten years (Round your answer to the nearest hundredth)?

Hint:

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

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

• import the number e from the math module.