Saturday, October 22, 2016

August 2016 Common Core Algebra II Regents Part 4

I don't usually post the answers to the Algebra II exam, but I've received enough requests that I've decided to post some of the open-ended questions.
Some questions were posted earlier here, here, here and here

August 2016, Algebra II Part 4

37. Seth’s parents gave him $5000 to invest for his 16th birthday. He is considering two investment options. Option A will pay him 4.5% interest compounded annually. Option B will pay him 4.6% compounded quarterly.
Write a function of option A and option B that calculates the value of each account after n years.

Seth plans to use the money after he graduates from college in 6 years. Determine how much more money option B will earn than option A to the nearest cent.

Algebraically determine, to the nearest tenth of a year, how long it would take for option B to double Seth’s initial investment.

Not an overly difficult Part 4 question, but it is a little involved.

For the first part, be sure to write a function and not just an expression. The initial amount is $5000, the rates are 0.045 and 0.046, respectively. Because the second one is quarterly, you have to divide the rate by 4 and multiple the exponent, n, by 4. (You would not be incorrect if you did this in the first function, using 1.)
A = 5000(1 + 0.045)n
B = 5000(1 + 0.046/4)4n

Use the two functions you just wrote, with n = 6. Write down both amounts and then subtract.
A = 5000(1 + 0.045)6 = 6511.30062424 = 6511.30
B = 5000(1 + 0.046/4)(4)(6) = 6578.86985121 = 6578.87
6578.87 - 6511.30 = 67.57

Doubling the $5000 investment means the future value will be $10000. Put this in the second function, and solve for n.
10000 = 5000(1 + 0.046/4)4n -- divide both sides by 5000
2 = (1 + 0.046/4)4n -- change the rational expression into a decimal
2 = (1.0115)4n -- solve using logs
4n = log1.01152 -- put this into your calculator
4n = 60.6196... -- divided by 4
n = 15.1549
In 15.2 years, Seth's investment will double.

End of exam

Any questions?

Thoughts, comments, concerns?

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