Algebra 2 Problems of the Day (Algebra 2/Trigonometry Regents, January 2011)

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, January 2011

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


28. Solve algebraically for x: 16^{2x + 3} = 64^{x + 2}

Answer:

Make the bases the same, and then compare the exponents.

Both 16 and 64 are is a powers of 2.

Rewrite the equation as 2^{(4)(2x + 3)} = 2^{(6)(x + 2)}

Now solve (4)(2x + 3) = (6)(x + 2)

(4)(2x + 3) = (6)(x + 2)

8x + 12 = 6x + 12

8x = 6x

2x = 0

x = 0

You can check your work if you don't believe 0 is the answer:

16³ = 4096, 64² = 4096

29.Find, to the nearest tenth of a degree, the angle whose measure is 2.5 radians

Answer:

There are 180 degrees in π radians. Set up a proportion and solve it:

180/π = x/2.5

πx = (2.5)(180)

x = (2.5)(180)/π

x = 143.23...

Approximate 143.2 degrees to the nearest tenth of a degree.

Since 180 degrees is approximately 3.141592... radians, this seems like a reasonable amount.

If you calculated the number of radians in 2.5 degrees (correctly), you would receive one credit.

30. For a given set of rectangles, the length is inversely proportional to the width. In one of these rectangles, the length is 12 and the width is 6. For this set of rectangles, calculate the width of a rectangle whose length is 9.

Answer:

Inversely proportional means that LW is constant, so L₁W₁ = L₂W₂

(12)(6) = (9)W₂

W₂ = 72/9 = 8

31. Evaluate

Answer:

You can do this the long way or the short way.

The long way:

n = 1, (1³ - 1) = 1 - 1 = 0

n = 2, (2³ - 1) = 8 - 1 = 7

n = 3, (3³ - 1) = 27 - 1 = 26

n = 4, (4³ - 1) = 64 - 1 = 63

n = 5, (5³ - 1) = 125 - 1 = 124

10 + 0 + 7 + 26 + 63 + 124 = 230

Quicker would be the know that 1³ + 2³ + 3³ + 4³ + 5³ = (1 + 2 + 3 + 4 + 5)².

So 10 + 15² - 5 = 10 + 225 - 5 = 230.

More to come. Comments and questions welcome.

More Regents problems.

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