After a brief hiatus, the Algebra 2 Problems of the Day are back. Hopefully, daily.

More Algebra 2 problems.

__January 2019, Part I__

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

*4. When a ball bounces, the heights of consecutive bounces form a geometric sequence. The height of the first bounce is
121 centimeters and the height of the third bounce is 64 centimeters. To the nearest centimeter, what is the height of the
fifth bounce?
(1) 25
(2) 34
(3) 36
(4) 42
*

**Answer: (2) 34**

We know a_{1} = 121 and a_{3} = 64.

The common ratio, r = 64 / a_{2} or a_{2} / 121

Then r^{2} = (64 / a_{2}) (a_{2} / 121) = 64 / 121

And r = SQRT(64/121) = 8/11

To get from a_{3} to a_{5}, you need to multiply by the common ratio two more times (or multiply by r^{2}).

64 * (8/11)*(8/11) = 33.851... = 34

*5. The solutions to the equation 5x ^{2} - 2x + 13 = 9 are
(1) 1/5 + SQRT(21)/5
(2) 1/5 + SQRT(19)/5 i
(3) 1/5 + SQRT(66)/5 i
(4) 1/5 + SQRT(66)/5
*

**Answer: (2) 1/5 + SQRT(19)/5 i**

If 5x

^{2}- 2x + 13 = 9

then 5x

^{2}- 2x + 4 = 0

If you graph this, you will see that there are no real roots, and you can eliminate (1) and (4).

Calculate the discriminate,

*= (-2)*

**b**^{2}- 4ac^{2}- 4(5)(4) = 4 - 80 = -76

SQRT(-76) = SQRT(-1 * 4 * 19) = 2

*i** SQRT(19), which elminates choice (3).

x = (-b __+__ SQRT (b^{2} - 4ac) ) / (2a)

x = ( -(-2) __+__ SQRT (-76) ) / (2*5)

x = ( 2 __+__ 2*i* SQRT (19)) / (10)

Split the fraction

x = 2/10 __+__ 2*i* SQRT (19) / 10

x = 1/5 __+__ SQRT(19)/5 *i*

*6. Julia deposits $2000 into a savings account that earns 4% interest
per year. The exponential function that models this savings account is y = 2000(1.04) ^{t}, where t is the time in years. Which equation correctly represents the amount of money in her savings account in terms of the monthly growth rate?
(1) y = 166.67(1.04)^{0.12t}
(2) y = 2000(1.01)^{t}
(3) y = 2000(1.0032737)^{12t}
(4) y = 166.67(1.0032737)^{t}
*

**Answer: (3) y = 2000((1.0032737) ^{12t}**

If you take the 12th root (1/12 power) of 1.04, you get 1.00327373978...

Conversely, if you raise 1.0032737 to the 12th power, you will get 1.039999... or 1.04.

Comments and questions welcome.

More Algebra 2 problems.

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