Question

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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 7% 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 at the annual interest rate is $32 million.

The arena should be financed by selling $________ million in bonds and borrowing $__________ million.

Answer 1

The arena should be financed by selling $320 million in bonds and borrowing $480 million.

Explanation:

Simple interest is calculated by multiplying the principal by the interest rate (in decimal form) and the time (in years).

Let x be the amount raised by selling bonds that pay an annual simple interest rate of 7%.

Therefore:

• Interest = 0.07x

As the total cost of the sports arena is $800 million, the amount raised from borrowing money from the insurance company at an annual simple interest rate of 2% is (800 - x). Therefore:

• Interest = 0.02(800 - x)

To determine whether the arena can be financed so that the annual interest is $32 million, equate the sum of the two expressions for interest to 32 and solve for x:

`\implies 0.07x+0.02(800-x)=32`

`\implies 0.07x+16-0.02x=32`

`\implies 0.05x+16=32`

`\implies 0.05x=16`

`\implies x=320`

As x is the amount raised by selling bonds, then the amount in bonds that should be sold is $320 million.

To calculate the amount to borrow, subtract the amount in bonds sold from $800 million:

`\implies 800-320=480`

Therefore, the amount to be borrowed from an insurance company is $480 million.

As both these amounts are less than the maximum limits set, they are both valid.

Answer 2

The arena should be financed by selling $ 320 million in bonds and borrowing $ 480 million.

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Work Shown:

• x = amount (in millions of dollars) from bonds at 7% interest rate.
• y = amount (in millions of dollars) borrowed at 2% interest rate.

Something like x = 27 means "27 million dollars of bonds were sold".

The variable x has the restriction of
`0 \le x \le 480` since $480 million is the ceiling for the bonds.

The variable y has the restriction of
`0 \le y \le 640` since $640 million is the ceiling for the money borrowed from the insurance company.

The two amounts, x and y, must add to 800 to represent the $800 million total that needs to be raised.

x+y = 800

y = 800-x

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Let's calculate the interest for each item. Use the simple interest formula.

Bonds at 7% interest rate:

i = P*r*t

i = x*0.07*1

i = 0.07x

Borrowed money at 2% interest rate:

i = P*r*t

i = y*0.02*1

i = 0.02y

Total annual interest = 0.07x+0.02y = 32

The 32 at the end represents the $32 million of interest payout. This amount of money is paid to the bondholders and to the insurance company, and is done so yearly.

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Apply substitution to solve for x.

0.07x+0.02y = 32

0.07x+0.02(800-x) = 32 ...... plug in y = 800-x

0.07x+16-0.02x = 32

0.05x+16 = 32

0.05x = 32-16

0.05x = 16

x = 16/(0.05)

x = 320

They should sell $320 million in bonds.

320 million = 320,000,000

Note how x = 320 is in the interval
`0 \le x \le 480`

y = 800-x

y = 800-320

y = 480

They should borrow $480 million at an interest rate of 2%

480 million = 480,000,000

Note how y = 480 is in the interval
`0 \le y \le 640`