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) i cant figure it out.

This question is answered by Merlynthewhizz: 
he scatter graph of the data is shown in the first picture below
The 'line of best fit' shows a negative gradient
Part A: The most likely coefficient is -0.98. If the coefficient is -1, then each point would be exactly on the straight line (which they are not as shown on the graph). The graph however still shows a strong negative coefficient. It can be seen from the close distance of each point from the line of best fit. So -0.5 and -0.02 is unlikely as they show weak negative correlation
Part B: Refer to the second picture to see the horizontal and vertical distance between day 15 and day 20
The horizontal distance is 5 unitsThe vertical distance is read between 61 and 71.5, hence it's 10.5The slope = Vertical distance ÷ Horizontal Distance = 10.5÷5 = 2.1The 'downhill' slope shows a negative gradient, hence the value of the slope is -2.1The value of the slope shows that the surface area of the lake shrinks by 2.1 for every one day
Part C: The data in the table represents the relation between two variables. Since one variable doesn't cause the change in the other variable, the data table represents correlation rather than a causation. 
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Answer

1 number

explanation

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

 

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.