Linear Modelling Worksheet

Linear models are among the simplest, and yet most useful, types of mathematical models. Being able to develop and apply a linear model is a necessary skill for any branch of science.

These questions are designed to help you develop the following skills: Sketching a linear graph Writing a linear equation Using a linear equation to make predictions Determining the gradient and intercepts of a linear equation Interpreting the gradient and intercepts in context

Water tank problem1. A full water tank has to be drained in order to be cleaned. After 10 minutes of draining, the tank still has 1300 litres in it. After 30 minutes, it has 900 litres in it. Assume that the volume of water in the tank is decreasing linearly with time. (a) What are the two variables for this situation? Which is the independent variable, and which the dependent variable?

(b) Sketch the graph showing the relationship between the two variables. Label your axes.

(c) What is the gradient of your graph? What does this number represent? What is the significance of the negative sign?

(d) What is the y-intercept of your graph? What does this number represent?

(e) What is the x-intercept of your graph? What does this number represent?

(f) What is the equation of your graph?

(g) How much water would be left in the tank after 45 minutes? (First, use your graph to answer this question. Then check your answer by using your equation.)

(h) How long will it take for the tank to be only one-quarter full? (First, use your graph to answer this question. Then check your answer by using your equation.)

Two water tanks problem2. Suppose there is a second (full) water tank beside the tank in question 1. This tank also has to be drained. After 10 minutes of draining, the second tank still has 1800 litres in it. After 30 minutes, it has 1200 litres in it. The taps for draining the two tanks were turned on at the same time. (a) On the same axes, draw the graph showing the relationship between the two variables for the second tank.

(b) What is the significance of the point of intersection of the two lines?

(c) Which of the two tanks will be empty first?

(d) What is the equation of the second graph?

(e) Solve the two equations simultaneously. Interpret your result.

(f) Find the x-intercepts of the two equations. Interpret these results?

Driving home problem3. As you drive home from work, the number of kilometres you are away from home depends on the number of minutes you have been driving. Assume the relationship between the two variables is linear. Suppose you are 11 km from home when you have been driving for 10 minutes, and you are 8 km from home when you have been driving for 15 minutes. (a) What are the two variables for this situation? Which is the independent variable, and which the dependent variable?

(b) Sketch the graph showing the relationship between the two variables. Label your axes.

(c) What is the gradient of your graph? What does this number represent? What is the significance of the negative sign?

(d) What is the y-intercept of your graph? What does this number represent?

(e) What is the x-intercept of your graph? What does this number represent?

(f) What is the equation of your graph?

(g) Predict your distance from home after driving for 25 minutes? (First, use your graph to answer this question. Then check your answer by using your equation.)

(h) When you were 7 km from home, for how many minutes had you been driving? (First, use your graph to answer this question. Then check your answer by using your equation.)

Chirping cricket problem4. It has been hypothesised that the rate at which crickets chirp is a linear function of temperature. At 15 C they make approximately 76 chirps per minute. At 47 C they make approximately 100 chirps per minute. (a) Write the linear equation expressing the chirping rate (C) in terms of temperature (T).

(b) Predict the chirping rate for 32 C.

(c) What is the probable temperature if you count a cricket chirping at 120 chirps per minute?

(d) Plot the graph of this linear model.

(e) What is the gradient of this model? What does this number represent?

(e) Find and interpret the y-intercept for this chirping model. Do you think the model is likely to be valid at this point?

(f) Find and interpret the x-intercept for this chirping model. Do you think the model is likely to be valid at this point?

ANSWERS

Water tank problem1. A full water tank has to be drained in order to be cleaned. After 10 minutes of draining, the tank still has 1300 litres in it. After 30 minutes, it has 900 litres in it. Assume that the volume of water in the tank is decreasing linearly with time. (a) What are the two variables for this situation? Which is the independent variable, and which the dependent variable?Independent variable: time (minutes)Dependent variable: volume of water in tank (litres)(b) Sketch the graph showing the relationship between the two variables. Label your axes.

(c) What is the gradient of your graph? What does this number represent? What is the significance of the negative sign?m = -20 litres/minute (The gradient has units because the variables have units.)This number represents the rate at which water is draining out of the tank. The negative sign indicates that the volume of water in the tank is decreasing. (d) What is the y-intercept of your graph? What does this number represent?y-intercept = 1500 litresThis represents the initial volume of water in the tank before the tap was turned on. (e) What is the x-intercept of your graph? What does this number represent?x-intercept = 75 minutesThis represents the time it takes to completely empty the tank. (f) What is the equation of your graph?Let V be the volume of water in the tank. Let t be the time. V = -20t + 1500(g) How much water would be left in the tank after 45 minutes? (First, use your graph to answer this question. Then check your answer by using your equation.)The graph indicates that the answer is 600 L. Using the equation:V = -20(45) + 1500 = -900 + 1500 = 600 L(h) How long will it take for the tank to be only one-quarter full? (First, use your graph to answer this question. Then check your answer by using your equation.) of 1500 = 475 litresThe graph indicates that the tank has 475 L after approximately 56 minutes. Using the equation:375 = -20t + 1500-1125 = -20tt = -1125 / -20 = 56.25 minutes

Two water tanks problem2. Suppose there is a second (full) water tank beside the tank in question 1. This tank also has to be drained. After 10 minutes of draining, the second tank still has 1800 litres in it. After 30 minutes, it has 1200 litres in it. The taps for draining the two tanks were turned on at the same time. (a) On the same axes, draw the graph showing the relationship between the two variables for the second tank.

(b) What is the significance of the point of intersection of the two lines?This is the time when the two tanks have the same volume.

(c) Which of the two tanks will be empty first? The second tank is empty after 70 minutes. The first isnt empty until 75 minutes.

(d) What is the equation of the second graph?V = 30t + 2100

(e) Solve the two equations simultaneously. Interpret your result.V = -20t + 1500 (1)V = 30t + 2100 (2)Substitute equation (2) into equation (1) for V. -30t + 2100 = -20t + 1500-10t = -600t = 60 minutesSubstitute t = 60 into either of the above equations (Ill choose (1)) to find the corresponding value for V. V = -20(60) + 1500 = 600 LThe solution is t = 60 minutes, V = 600 L. Solving two equations simultaneously gives the point of intersection of the two graphs.

(f) Find the x-intercepts of the two equations. Interpret these results?From the graphs:

x-intercept for first tank equation = 75 minx-intercept for second tank equation = 70 minThese are the times taken for the two tanks to empty.

(The x-intercepts can also be found by substituting y = 0 into the equations. )

Gary Pocock4/06/201411