This exam was adminstered in August 2024 .

### August 2024 Algebra 2, Part II

Each correct answer is worth up to 2 credits. Partial credit can be given. Work must be shown or explained.

**25.** * On the axes below, graph y = 3.2(1.8) ^{2}
.
*

**Answer: **

Put the equation in your graphing calculator and check the Table of Values.

Points on the graph include: (0,3.2), (1,5.76), (2,10.368), (3,18.6624), (-1, 1.78), (-2,.99), (-3,.55), (-4,.3), (-5,.16). Approxminate these points.

**26.** * Is x + 3 a factor of 7x ^{3} + 27x^{2} + 9x - 27?
*

Justify your answer.

**Answer: **

You can show if it is a factor or not by doing doing polynomial division. However, an easier method is this: if (x + 3) is a factor, then the polynomial will equal zero when x = -3.

7(-3)^{3} + 27(-3)^{2} + 9(-3) - 27 = 0, so (x + 3) is a factor of the polynomial.

**27.** * Over the set of integers, factor the expression 2x ^{4} - 10x^{3} + 3x^{2} - 15x completely.
*

**Answer: **

Factor completely means that there will be multiple steps. Over the set of integers means that you don't have to worry about imaginary numbers.

2x^{4} - 10x^{3} + 3x^{2} - 15x

x(2x^{3} - 10x^{2} + 3x - 15)

x(2x^{2}(x - 5) + 3(x - 5))

x(2x^{2} + 3)(x - 5)

**28.** *The monthly unemployment rate of towns in the United States is approximately normally
distributed with a mean rate of 5.2% and a standard deviation of 1.6%. Determine the percentage of towns, to the nearest integer, that have a monthly unemployment rate greater than 6%.
*

**Answer: **

Use the "normalcdf" function on your graphing calculator.

Use the command normalcdf(6,100,5.2,1.6) to find the percentage between 6% and 100% ("greater than 6%), mean 5.2%, standard deviation 1.6%.

The result should be .3085, which is .31, or 31%.

**29.** * The function d(t) = 2cos(π/6 t) + 5 models the water depth, in feet, at a location in a bay, t hours
since the last high tide. Determine the minimum water depth of the location, in feet, and justify
your answer.
*

**Answer: **

The midline is 5 feet (from the +5). The amplitude is 2 feet (from 2cos). This means that the water depth varies between 3 feet and 7 feet.

Therefore, the minimum water depth is 3 feet.

Remember to justify it. This is easy enough to do in your head. Do NOT just write 3 feet, or 5 - 2 = 3. Indicate where the 5 and 2 came from and what they stand for.

**30.** * A brewed cup of coffee contains 130 mg of caffeine. The half-life of caffeine in the bloodstream
is 5.5 hours. Write a function, C(t) to represent the amount of caffeine in the bloodstream
t hours after drinking one cup of coffee
*

**Answer: **

The function is expoonential, with 130 as the initial amount and 1/2 as the rate (half-life). It takes 5.5 hours for 1/2 to leave the bloodstream, but t is measured in hours, so we need to divide t by 5.5.

C(t) = 130(1/2)^{t/5.5}

**31.** * Markus is a long-distance walker. In one race, he walked 55 miles in t hours and in another race
walked 65 miles in t + 3 hours. His rates are shown in the equations below.
*

*r = 55/t ; r = 65 / (t + 3)*

*Markus walked at an equivalent rate, r, for each race. Determine the number of hours that each of
the two races took.
*

**Answer: **

Write a proportion and solve it.

55/t = 65/(t + 3)

65t = 55(t + 3)

65t = 55t + 165

10t = 165

t = 16.5

The first race took 16.5 hours, and the second race took 19.5 hours.

**32.** * Solve the equation x ^{2} + 3x + 11 = 0 algebraically. Express the answer in a + bi form.
*

**Answer: **

Use the Quadratic Formula to solve this.

x = (-b __+__ √(b^{2} - 4ac) / (2a)

x = (-3 __+__ √((3)^{2} - 4(1)(11)) / (2(1))

x = (-3 __+__ √(9 - 44)) / (2)

x = (-3 __+__ √(-35)) / (2)

x = -3/2 __+__ √(35)/2 i

Note that you MUST separate the expression into two fraction because the question states a + bi form.

**End of Part II**

How did you do?

Questions, comments and corrections welcome.

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