Any help at all is deeply appreciated. I am trying to figure out the formulas to use for the following problems:
1. Suppose that in a monopoly market the total cost per week of producing a high tech product is given by C(x) =3600+100x+2x^2. Suppose further that the weekly demand function for this product is p=500-2x. Find the number of units that will give the break even points for the product. Find the number of units that will give maximum profit.
I think the C(x) is suppose to = the p but I can't seem to get that to work.
2. Bond Pricing: A 20-year corporate bond has a maturity value of $25, 000 and coupons are paid at 5% at the end of each year. If an investor wants to earn a yield of 7% compounded semiannually, what should she or he pay for this bond?
3. A couple is saving for their child's college. They decided they would like to have $50,000 in 18 years. If they can earn 4.5% compounded semiannually, how much should they deposit at the beginning of each period?
Hmmm. Too bad you didn't show your work...
revenue = price * quantity.
breakeven is when cost = revenue, so
3600+100x+2x^2 = x(500-2x)
x = 10,90
maximum profit will be at the vertex of p = r-c, halfway between the roots: x=50
Sure! I'll explain how to approach each of these problems and provide the formulas that can be used:
1. Monopoly Market:
To find the break-even point, you need to set the cost function C(x) equal to the price function p(x) and solve for x. In this case, we have:
C(x) = 3600 + 100x + 2x^2
p(x) = 500 - 2x
Setting C(x) equal to p(x), we get:
3600 + 100x + 2x^2 = 500 - 2x
By rearranging the equation and simplifying, we have:
2x^2 + 102x - 500 = 0
You can solve this quadratic equation using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 2, b = 102, and c = -500. Plug in these values to find the solutions for x, which will give you the break-even points.
To find the number of units that will give maximum profit, you need to determine the revenue function and the profit function. The revenue function is given by R(x) = p(x) * x, and the profit function is given by P(x) = R(x) - C(x).
Once you have the profit function, you can find the maximum profit by taking the derivative of P(x) with respect to x, setting it equal to zero, and solving for x. This will give you the value of x that maximizes the profit.
2. Bond Pricing:
To determine the price of the bond, you can use the present value formula:
Price = Coupon * [1 - (1 + r/c)^(-n * c)] / (r/c) + M * (1 + r/c)^(-n * c)
In this formula:
- Coupon represents the annual coupon payment, which is 5% of the maturity value ($25,000) in this case.
- r is the annual yield, which is 7% in this case.
- c is the number of compounding periods per year, which is semiannually (2 in this case).
- n is the number of years, which is 20 in this case.
- M is the maturity value, which is $25,000 in this case.
Plug in these values into the formula, and you'll get the price that the investor should pay for the bond.
3. College Savings:
To determine how much the couple should deposit at the beginning of each period, you can use the present value of an annuity formula:
Deposit = A * [1 - (1 + r/c)^(-n * c)] / (r/c)
In this formula:
- A is the future value they want to have ($50,000 in this case).
- r is the annual interest rate, which is 4.5% in this case.
- c is the number of compounding periods per year, which is semiannually (2 in this case).
- n is the number of years, which is 18 in this case.
By plugging in the values into the formula, you'll find the amount they need to deposit at the beginning of each period to reach their savings goal.
I hope this helps! Let me know if you have any further questions.