Sunday, November 25, 2018

Algebra 2 Problems of the Day

Daily Algebra 2 questions and answers.

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June 2017, Part I

All Questions in Part I are worth 2 credits. No work need be shown. No partial credit.

22. Mallory wants to buy a new window air conditioning unit. The cost for the unit is $329.99. If she plans to run the unit three months out of the year for an annual operating cost of $108.78, which function models the cost per year over the lifetime of the unit, C(n), in terms of the number of years, n, that she owns the air conditioner?
(1) C(n) = 329.99 + 108.78n
(2) C(n) = 329.99 + 326.34n
(3) C(n) = (329.99 + 108.78n) / n
(4) C(n) = (329.99 + 326.34n) / n

Answer: (3) C(n) = (329.99 + 108.78n) / n
Choice (1) is the total cost to operate the unit over its lifetime. To find the yearly cost, divide that amount by the number of years, n.
Alternatively, if it costs $108.78 to operate per year (and then the cost is 108.78n to operate for n years). The cost of the unit is paid once (initial value) and divided by the number of years the unit is owned to get a "per year" cost, which is 329.99/n.
So the yearly cost is: 329.99/n + 108.78, which is equivalent to (329.99 + 108.78n) / n.

23. The expression (-3x2 - 5x + 2) / (x3 + 2x2)

can be rewritten as

Answer: (4) -3x-1 + x-2
As shown in the image below. Factor both the numerator and the denominator.
The numerator is (-1)(3x2 + 5x - 2), which factors into (-1)(3x - 1)(x + 2).
The denominator is (x2)(x + 2).
Eliminate the common factor of x + 2.
At this point, choice (2) looks so close to being correct, but it has a numerator of -3x - 1, instead of -3x + 1.
Use the division rule and reduce the exponents in the numerator by 2, remembering that the last term is actually 1x0.
This gives us choice (4).

24.Jasmine decides to put $100 in a savings account each month. The account pays 3% annual interest, compounded monthly. How much money, S, will Jasmine have after one year?

Answer: (2) S = (100 - 100(1.0025)12)/(1 - 1.0025)
When compounding interest, if $100 were deposited initially, with no additional deposits, then choice (3) would have been correct. The interest rat of 3%, or 0.03, is spread out over 12 months. In other words, 0.03/12 = 0.0025 is the monthly interest, raised to the power of 12 for the entire year. You can eliminate choices (1) and (4), both of which use 1.03.
Because additional deposits are being made monthly, choice (3) is incorrect. The more complicated formula shown in choice (2) comes from the Geometric Sequence formula in the reference table at the back of the booklet:

Sn = (a1 - a1rn) / (1 - r), where r =/= 1.
In this case, a1 = 100, and r = 1.0025.

Comments and questions welcome.

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