Question

The table below shows the surface area y, in square feet, of a shrinking lake in x days:


Time (x)
(days) 10 20 30 40
Surface area (y)
(square feet) 100 90 80 70


Part A: What is the most likely value of the correlation coefficient of the data in the table? Based on the correlation coefficient, describe the relationship between time and surface area of the lake. [Choose the value of the correlation coefficient from −1, −0.99, −0.5, −0.02.] (4 points)

Part B: What is the value of the slope of the graph of surface area versus time between 20 and 30 days, and what does the slope represent? (3 points)

Part C: Does the data in the table represent correlation or causation? Explain your answer. (3 points)

Use the functions h(x) = 5x − 2 and t(x) = 4x + 6 to complete the function operations listed below.

Part A: Find (h + t)(x). Show your work. (3 points)

Part B: Find (h ⋅ t)(x). Show your work. (3 points)

Part C: Find h[t(x)]. Show your work. (4 points)

Part A: Factor 3x2y2 − 2xy2 − 8y2. Show your work. (4 points)

Part B: Factor x2 + 10x + 25. Show your work. (3 points)

Part C: Factor x2 − 36. Show your work. (3 points)

A quadratic equation is shown below:

4x2 − 12x + 10 = 0

Part A: Describe the solution(s) to the equation by just determining the radicand. Show your work. (5 points)

Part B: Solve 2x2 − 13x + 21 = 0 by using an appropriate method. Show the steps of your work, and explain why you chose the method used. (5 points)

Answer 1

1A) The points are all on the line y=110-x, so the correlation coefficient is -1.

1B) The slope of the graph everywhere is -1 (square feet per day). This represents the change in area each day (-1 ft²).

1C) The data represents correlation. There is no reason to believe the change in area is caused by the passage of time. Rather, we expect the change to be caused by some process that removes water from the lake. In fact, we expect surface area to increase with time if water is being added (as by rain, for example).


2A) (h+t)(x) = h(x) +t(x) = (5x -2) +(4x +6) = 9x +4

2B) (h·t)(x) = h(x)·t(x) = (5x -2)·(4x +6) = 20x² +22x -12

2C) h(t(x)) = h(4x +6) = 5(4x +6) -2 = 20x +28


3A) 3x²y² -2xy² -8y² = y²(3x² -2x -8) = y²(x -2)(3x +4)
  we made use of the factoring -24 = -6·4 to form factors (3x-6)/3·(3x+4)

3B) x² +10x +25 = (x +5)² . . . . . matches the form for the square of a binomial

3C) x² -36 = (x -6)(x +6) . . . . . . matches the form for the difference of squares


4A) The discriminant is b²-4ac = (-12)² -4(4)(10) = 144 -160 = -16. The radicand is negative, so the solutions will be complex.

4B) A graph shows the solutions to be x ∈ {3, 3.5}.
  The "steps of the method" consisted of typing the expression into a graphing calculator and highlighting the x-intercepts. I chose this method as it requires the least amount of work.