Worksheet - linear modeling

8/10/2019 Worksheet - linear modeling

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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 problem

1. 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 volumeof 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.


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(c) What is the gradient of your graph? (Dont forgetthe units.) 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?



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Two water tanks problem

2. 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 variablesfor 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?


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Driving home problem

3. As you drive home from work, your distance 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, andwhich the dependent variable?


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

the negative sign?




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(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?



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Chirping cricket problem

4. 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 31 C they make approximately 100 chirps

per minute.

(a) Plot the graph of this linear model.

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

(c) Find and interpret the y-intercept for this chirping model. Do you think the model is likely to be

valid at this point?

(d) Write the linear equation expressing the chirping rate (C) in terms of temperature (T).


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(e) Predict the chirping rate for 32 C. First use your graph, then your equation.

(f) What is the probable temperature if you count a cricket chirping at 120 chirps per minute? Firstuse your graph, then your equation.

(g) Find and interpret the x-intercept for this chirping model. Do you think the model is likely to be



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Skydiver problem

5. A skydiver jumps from a plane at 1200 metres and immediately deploys her parachute. Information

about her descent is recorded below.

Time (s) Height (m)0 110015 995

43 79980 540130 190

(a) Plot these data on the grid below.

(b) Does a linear model seem appropriate for these data? If so, draw a line though the data points.

(c) What is the gradient of the line? Interpret this value.

(d) What is the equation of the line?

(e) Use your graph to determine how long it takes for the skydiver to reach the ground.


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(f) Use your equation to confirm the answer you found in par (e).

(g) A second skydiver jumps out of the plane 15 seconds after the first skydiver. He also deploys his

parachute immediately. Information about his descent are shown below.

Time (s) Height (m)0 -15 110043 86280 547.5130 122.5

Plot these data on the same grid, and draw the line through the points.

(h) Find the rate at which the second skydiver is falling.

(i) Will the second skydiver overtake the first skydiver before reaching the ground? If so, after how

many seconds? And at what height?

(j) What is the time interval between the landings of the two skydivers?


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ANSWERS

Water tank problem

1. 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)


(c) What is the gradient of your graph? (Dont forgetthe units.)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 negativesign indicates that the volume of water in the tank is decreasing.


y-intercept = 1500 litres

This represents the initial volume of water in the tank before the tap was turned on.

x

y

0 10 20 30 40 50 60 70 80

200

400

600

800

1000

1200

1400

1600

1800

(10, 1300)

(30, 900)


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x-intercept = 75 minutes

This represents the time it takes to completely empty the tank.


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?


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?


of 1500 = 475 litres

The graph indicates that the tank has 475 L after approximately 56 minutes.

Using the equation:

375 = -20t + 1500

-1125 = -20t

t = -1125 / -20 = 56.25 minutes


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Two water tanks problem

2. 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 variablesfor 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

x

y

0 10 20 30 40 50 60 70 80

500

1000

1500

2000


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(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 = -600

t = 60 minutes

Substitute t = 60 into either of the above equations (Ill choose (1)) to find the

corresponding value for V.

V = -20(60) + 1500 = 600 L

The 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 min

x-intercept for second tank equation = 70 min

These are the times taken for the two tanks to empty.

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


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Driving home problem

3. As you drive home from work, your distance 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, andwhich the dependent variable?

Distance from home dependent variable

Driving time independent variable


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

the negative sign?

For every 1 minute of passing time, the model predicts that the distance from home

decreases by 0.6 km.

The negative sign indicates that the y-value (distance from home) is decreasing as the x-

value (time) increases.


y-intercept = 17 km.

This is the initial distance from home i.e. the distance between work and home.


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From graph: x-intercept = 28 min (approx.)

This is the time taken to travel from work to home.


Equation: y = -0.6x + 17

(g) Predict your distance from home after driving for 25 minutes?


From graph: x = 25 min, y = 2 km (approx.)

From equation: y = -0.6(25) + 17 = 2 km

(h) When you were 7 km from home, for how many minutes had you been driving?


From graph: y = 7 km, x = 16.5 min (approx.)

From equation:

7 = -0.6x + 17

-10 = -0.6x

x = -10/-0.6

x = 16.7 min


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Chirping cricket problem

4. 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 31 C they make approximately 100 chirps

per minute.

(a) Plot the graph of this linear model.

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

For every 1 degree increase in temperature, the model predicts that the chirp rate will

increase by 1.5 chirps per minute.

(c) Find and interpret the y-intercept for this chirping model. Do you think the model is likely to be


From the graph: y-intercept = 54 chirps per minute (approx.)

Model is unlikely to valid here. Doubtful if crickets chirp at all at that temperature.

(d) Write the linear equation expressing the chirping rate (C) in terms of temperature (T).

y = 1.5x + c

To find c, substitute a known point into this equation and solve. Ill choose (15, 76)


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76 = 1.5(15) + c

76 = 22.5 + c

c = 76 22.5

c = 53.5

Equation: y = 1.5x = 53.5

(e) Predict the chirping rate for 32 C. First use your graph, then your equation.

From graph: x = 32, y = 102 chirps per min (approx.)

From equation: y = 1.5(32) + 53.5 = 48 + 53.5 = 101.5 chirps per min

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

use your graph, then your equation.

From graph: y = 120, x = 44 C (approx.)From equation:

120 = 1.5x + 53.5

66.5 = 1.5x

x = 66.5/1.5 = 44.3 C

(g) Find and interpret the x-intercept for this chirping model. Do you think the model is likely to be


Use equation, as graph doesnt extend that far.Set y = 0.

0 = 1.5x + 53.5

1.5x = -53.5

x = -53.5/1.5 = -35.7 C

Obviously the model isnt valid at such low temperatures.


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Skydiver problem

5. A skydiver jumps from a plane at 1200 metres and immediately deploys her parachute. Information

about her descent is recorded below.

Time (s) Height (m)0 110015 995

43 79980 540130 190

(a) Plot these data on the grid below.

(b) Does a linear model seem appropriate for these data? If so, draw a line though the data points. Yes.

(c) What is the gradient of the line? Interpret this value.

Choose two convenient points:

The skydiver is falling at a rate of 7 metres per second.


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(d) What is the equation of the line?

y = -7x + 1100

(e) Use your graph to determine how long it takes for the skydiver to reach the ground.

S = 155 seconds (approx.)

(f) Use your equation to confirm the answer you found in par (e).

Substitute y = 0 into the equation:

0 = -7x + 1100

0.7x = 1100

x = 1100/7 = 157 seconds

(g) A second skydiver jumps out of the plane 15 seconds after the first skydiver. He also deploys his

parachute immediately. Information about his descent are shown below.

Time (s) Height (m)0 -15 110043 86280 547.5130 122.5

Plot these data on the same grid, and draw the line through the points.


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(h) Find the rate at which the second skydiver is falling.

(i) Will the second skydiver overtake the first skydiver before reaching the ground? If so, after how

many seconds? And at what height?

From the graph:

2ndskydiver will overtake 1stskydiver at t = 80 s (approx). That is, 80 s after the

1stskydiver jumped. This occurs at around 550 m.

(j) What is the time interval between the landings of the two skydivers?

From the graph, the 2ndskydiver will land after approximately 143 s. (x-

intercept)

The 1stskydiver will land after approximately 157 s. (x-intercept)

Time interval = 157 143 = 14 s (approx)

Worksheet - linear modeling

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