Algebra 2 Problems of the Day (Algebra 2 Regents, June 2012)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

Algebra 2/Trigonometry Regents, June 2012

Part II: Each correct answer will receive 2 credits. Partial credit is possible.


28. Determine the value of n in simplest form.

i¹³ + i¹⁸ + i³¹ + n = 0

Answer:

i² = -1 and i⁴ = 1, then i⁶ = -1 and i⁸ = 1, etc.

i¹³ + i¹⁸ + i³¹ + n = 0
(i¹²)(i) + i¹⁸ + (i³⁰)(i) + n = 0
(1)(i) + (-1) + (-1)(i) + n = 0
-1 + n = 0
n = 1

29. The formula for continuously compounded interest is A = Pe^rt, where A is the amount of money in the account, P is the initial investment, r is the interest rate, and t is the time in years.

Using the formula, determine, to the nearest dollar, the amount in the account after 8 years if $750 is invested at an annual rate of 3%.

Answer:

Set up the equation and use your calculator.

A = Pe^rt
A = (750)e^(0.03)(8) = 953.436863
A = $953

30. Express cos θ (sec θ - cos θ), in terms of sin θ.

Answer:

Trigometry identities that will be helpful: sec θ = 1 / cos θ, and sin² θ + cos² θ = 1.

cos θ (sec θ - cos θ)
= cos θ ((1 / cos θ) - cos θ)
= 1 - cos² θ
= sin² θ

31. A cup of soup is left on a countertop to cool. The table below gives the temperatures, in degrees Fahrenheit, of the soup recorded over a 10-minute period.

Write an exponential regression equation for the data, rounding all values to the nearest thousandth.

Answer:

Put the data in lists and run an exponential regression. You'll get the following equation:

y = 180.377(0.954)^x

This is one of the times where there is no real work to show because it is assumed that you did everything in the calculator. Likewise, there is pretty much no way that you could have gotten the correct answer without using the calculator.

Because of this, there are only a few ways to get 1 credit, instead of two: not rounding correctly or swapping the digits (transcription error), for sure; writing a linear regression equation; or writing a correct exponential expression, instead of an equation.

That is to say, "180.377(0.954)^x" is worth 1 credit, as is "y = 180.377(0.95)^x". However, "180.377(0.95)^x" by itself is worth 0 credits.

More to come. Comments and questions welcome.

More Regents problems.

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