Modeling with linear equations -Stuck whether this is starting correctly and where to go.

Question

Number of rabbits in an area depends on the number of wolves as follows:

Wolves Rabbits

40 8000

44 7000

48 6000

A) find a linear equation showing how the number of rabbits depends on the wolves.

B) explain the meaning of the rate of change found in the context of the situation.

C) interpret the meaning of the constant term in your equation.

My A answer: I believe this starts as y=mx=b. So, 8000 = -1000(x) + b (this seems like I am way off)

My B answer: Would this be that as wolves increase by 4, rabbits decrease by 1000?

My C answer: I don't have the equation down but the constant is based off the beginning population?

Answer 1

The slope, using rise/run on 1st and last data points: Chg in rabbits/chg in wolves is -2000/8 = - 250.

y = mx + b -> Using one of the data pairs: 8000 = -250(40) + b -> 8000 = -10000 + b, -> b = 8000 + 10000 = 18000.

Eqn is y = -250x + 18000.

Chk @ 48 wolves; y = -250(48) + 18000 -> y = -12000 + 18000 = 6000 (OK)

Your answer to B) is correct. 4 more wolves means 1000 fewer rabbits, hence the slope of -1000/4 = -250.

C) The constant term b is the y-intercept of the graph of rabbits vs. wolves. With no wolves at all, there would be 18000 rabbits (y = 18000 @ x = 0).

Answer 2

I edited my original response. Peter R. correctly noted an error in my slope calculation. I divided -2000 by 48, not the 28 that was calculated for the run. I've revised the slope number and corrected results. Thanks, Peter R.

My A answer: I believe this starts as y=mx=b. So, 8000 = -1000(x) + b (this seems like I am way off)

I'm not sure how you arrived at 8000 = -1000(x) + b

Yes, look for an equation of the form

y = mx + b

m, the slope, can be determined by the Rise/Run of a straight line. Pick any two points. I'll choose (20, 8000) and (48, 6000).

Rise (6000-8000) = -2000

Run = (48 - 20) = 28

Slope = Rise/Run = -2000/28 or -71.429

This leads us to y = -71.429x + b

To find b, use one of the points in this equation and solve for b:

y = -71.429x + b for (20,8000)

8000 = -71.492*(20) + b

8000 = -1428.57+ b

b = 9428.57

The linear equation is y = --71.429x +9428.57

B) explain the meaning of the rate of change found in the context of the situation.

[My B answer: Would this be that as wolves increase by 4, rabbits decrease by 1000?] No

This rate of change tells us that the poplution of rabbits decreases by -71.429 for each additional wolf. 4 additional wolves would reduce the rabbitt count by 4*(-71.429) or -286.71 , rounded to -287 since -0.71 of a rabbitt will get to -1 very quickly.

C) interpret the meaning of the constant term in your equation.

[My C answer: I don't have the equation down but the constant is based off the beginning population?} Yes, the constant, 8833.33 is the starting rabbitt population before any wolves were present (x=0). By definition, the y-intercept (b) is the value of y when x=0 (no wolves). The line shows imaginary points when x<0, since it is difficult to have fewer than no wolves in this universe.

Answer 3

Start with two-point form. You can use any two of the three.

(W - 40) / (R - 8000) = (44 - 40) / (7000 - 8000)

Conververt to R = mW + b