SOLUTION: 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% annuall

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Question 1092819: 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 $21 million. (If there is no solution, enter NO SOLUTION.)
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
there is a feasible solution.

that solution is when x = 250 and y = 550

x represents the number of dollars in bonds that require payment of 4% interest.

y represents the number of dollars in loans from an insurance company that require payment of 2% interest.

the total number of dollars raised is x + y = 250 + 550 = 800 million.

the interest requirement on the number of dollars raised through issuance of bonds is 250 * .04 = 10 million.

the interest requirement on the number of dollars raised through the insurance company is 550 * .02 = 11 million.

the total interest requirement is 21 million.

this was found by finding the intersection of the following equations:

x + y = 800

.04x + .02y = 21

these equations are solved simultaneously.

we'll solve by substitution.

in the first equation, solve for y to get y = 800 - x.

in the second equation, replace y with 800 - x to get .04 * x + .02 * (800 - x) = 21

simplify this second equaton to get .04 * x + .02 * 800 - .02 * x = 21

simplify further and combine like terms to get .02 * x + 16 = 21

subtract 16 from both sides of the equation to get .02 * x = 5

solve for x to get x = 5 / .02 = 250.

since x + y = 800 and x = 250, then y has to be equal to 550.

your intersection is when x = 250 and y = 550

the next step is to see if this meets the requirements of the problem.

your constraints are that x <= 480 and y <= 640.

these constraints are met.

x + y = 800
.04x + .02y = 21

these requirements are met.

another constraint is that x and y both have be greater than or equal to 0.

these constraint have also been met.

graphically, your solution looks like this:

the point (250,550) is at the intersection of the lines of the equations of x + y = 800 and .04x + .02y = 21.

you can see that x-coordinate of 250 is smaller than x = 480 and you can see that the y-coordinate of 550 is smaller than y = 640.

you can also see that these values are both greater than 0.

therefore, the solution is feasible because you have an intersection of the required equations and that solution is in the feasible region.