Algebra 2 Problems of the Day (Algebra 2 Regents, August 2024 Part III)

This exam was adminstered in August 2024 .

August 2024 Algebra 2, Part III

Each correct answer is worth up to 4 credits. Partial credit can be given. Work must be shown or explained.

33. The population of China, in millions, can be modeled by the function P(x) = 316.93e^0.0133x , where x is the number of years since 1900. The population of India since 1900 is summarized in the table below:

Which country’s population had a greater average rate of change between 1950 and 2020? Justify your answer.

Answer:

Find the rate of change for India by subtracting the population in 1950 from the population in 2020, and divide it by 70 (years). Find the change for China by evaluating the function for x = 50 and x = 120 (years since 1900) and divide that by 70. Which is bigger?

Note that you can solve the problem without dividing each number by 70, but they specifically asked about the rates, so divide by 70.

India: (1380 - 376.3) / 70 = 1003.7 / 70

China: P(50) = 316.93e^0.0133(50) = 616.267
P(120) = 316.93e^0.0133(120) = 1563.498
(P(120) - P(50)) / 70 = (1563.498 - 616.267) / 70 = 947.231 / 7

India had a greater rate of change because 1003.7 / 70 > 947.231 / 70.

34. In a packaging plant, a machine packs boxes with jars. The machine’s manufacturer states that a box is packed, on average, every 42 seconds. To test that claim, the packaging plant randomly selects a sample of 10 boxes and finds the sample mean to be 49.8 seconds.

The company ran a simulation of 1000 trials based on the manufacturer's claim. The approximately normal results are shown below.

Based on the simulation, determine an interval containing the middle 95% of plausible mean times. Round your answer to the nearest hundredth.

Is the time 49.8 seconds unusual? Use statistical evidence to justify your answer.

Answer:

The middle 95% is the mean minus 2 times the standard deviation to the mean plus 2 times the standard deviation.

42.029 - 6.21 = 35.819, 42.029 + 6.21 = 48.239. (35.82, 48.24) to the nearest hundredth.

The value 49.8 is unusual because it falls outside the middle 95%. (It is greater than 48.24.)

35. Consider the function f(x) = 2^x.
Is f(x) an even function? Justify your answer.

Write an equation for g(x), the function that results after f(x) is shifted up 5 units.

Write an equation for h(x), the inverse of g(x).

Answer:

The function f(x) is not an even function because f(x) =/= f(-x). For example f(1) = 2, f(-1) = 1/2.

g(x) = f(x) + 5 = 2^x + 5

Note: I checked the official response guide and "f(x) + 5" was acceptable as an answer.

To find the inverse, replace x with y, and set the function equal to y. Then solve for x.

x = 2^y + 5
x - 5 = 2^y
log₂ (x - 5) = y

h(x) = log₂ (x - 5)

36. Solve the system of equations shown below algebraically:

(x - 4)² + (y - 1)² = 9
x - y = 6

Answer:

Find the two points on the line that intersect the circle.

Solve for x and substitute in the circle equation.

x - y = 6
x = y + 6

(x - 4)² + (y - 1)² = 9
(y + 6 - 4)² + (y - 1)² = 9
(y + 2)² + (y - 1)² = 9
y² + 4y + 4 + y² - 2y + 1 = 9
2y² + 2y + 5 = 9
2y² + 2y - 4 = 0
y² + y - 2 = 0
(y + 2)(y - 1) = 0
y + 2 = 0 or y - 1 = 0
y = -2 or y = 1

If y = -2, x - (-2) = 6, then x = 4. (4, -2)

If y = 1, x - 1 = 6, then x = 7. (7, 1)

End of Part III

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