Question

A city government has approved the construction of an $800 million sports arena. Up to $480 million will be raised by selling bonds that pay simple interest at a rate of 4% annually. The remaining amount (up to $640 million) will be obtained by borrowing money from an insurance company at a simple interest rate of 2%. Determine whether the arena can be financed so that the annual interest is $22 million. (If there is no solution, enter NO SOLUTION.)

Answer 1

borrowing structure:

500 insruance at 2%

300 bonds at 4%

weighted average rate: 2.75%

Step-by-step explanation:

we will calculate which rate generates an interes texpense of 22 million per year:

22,000,000 / 800,000,000 = 0,0025 = 2.75%

we need a weighted average rate for 2.75%

we construct an equation system:

`(B_i * 0.02+B_b* 0.04)/(800) =0.0275\\B_i + B_b = 800\\`

insurnace company = 800 - bonds

we now solve for bond quantity:

`((800-B_b) * 0.02+B_b* 0.04)/(800) =0.0275`

`800 * 0.02 - 0.02B + 0.04B = 0.0275 * 800`

`0.02B_b = 0.0275 * 800 - 800 * 0.02`

B = 6/0.02 = 300

now we got bonds borring we calcualte for insurance borrowing:

800 - 300 = 500 million

we check:

300,000,000 x 0.04 = 12,000,000 interest

500,000,000 x 0.02 = 10,000,000 interest

total 22,000,000 interest