Question

On your birthday your rich uncle gave you $19000. You would like to invest at least $9500 of the money in municipal bonds yielding 3% and no more than $4000 in Treasury bills yielding 6%. How much should be placed in each investment in order to maximize the interest earned in one year? Assume simple interest applies. Let x represent the amount of money in municipal bonds and y represent the amount of money in Treasury bills.

Answer 1

To maximize the interest earned in one year, set up the constraints and objective function, and use linear programming to determine the optimal investments.

Step-by-step explanation:

To maximize the interest earned in one year, you need to set up a system of constraints and an objective function. Let x represent the amount of money invested in municipal bonds and y represent the amount invested in Treasury bills.

Based on the given information, we have the following constraints:

• x ≥ $9500
• y ≥ 0
• x + y ≤ $19000
• y ≤ $4000

The objective function is to maximize the interest earned, which is given by: I = 0.03x + 0.06y

Using these constraints and the objective function, you can solve the linear programming problem using graphical or algebraic methods to determine the optimal amounts to invest in municipal bonds and Treasury bills.

Answer 2

To maximize interest, invest $9,500 in municipal bonds at 3% and $4,000 in Treasury bills at 6%, achieving a total interest of $525. When market interest rates are higher than the bond's rate, as with the local water company bond, it will sell for less than face value; in the given scenario, the bond's price would not exceed $964.

Step-by-step explanation:

To maximize the interest earned in one year, the student must invest $9,500 in municipal bonds at a 3% return and the remaining $4,000 in Treasury bills at a 6% return. This is determined based on the conditions that at least $9,500 should be invested in municipal bonds and no more than $4,000 in Treasury bills.

To illustrate, if $x$ represents the amount in municipal bonds and $y$ represents the amount in Treasury bills:

• 
  
• If $x = $9,500 and $y = $4,000, then the total interest is $(0.03 \times 9500) + (0.06 \times 4000)$
• 
  
• Total Interest = $285 + $240 = $525

Any amount less than $9,500 in municipal bonds or more than $4,000 in Treasury bills will result in less total interest earned over the year.

When it comes to buying the local water company bond at a 6% rate when the market interest rate is 9%, investors would expect to pay less than the bond's face value of $10,000 due to the higher prevailing market interest rates.

If the expected payments from the bond one year from now are $1,080, and the market interest rate is 12%, the bond's price when its interest rate is less than the market interest rate would not exceed $964. By using the formula:

• 
  
• $964 (1 + 0.12) = $1,080

This implies that the current value of the bond's expected payments must be equal to the amount that could be earned from an alternative investment at the new market rate of 12%. Thus, the maximum that should be paid for this bond is $964.