Maths Methods task.docx

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Grids on pages 5 and 9 can be used to sketch graphs.

PART 1

SEND: Modelling Task (Graded)

PROBLEM 1 TheRichter scale(2+ 1 + 1 +2=6marks)

The Richter scale is used to compare the intensities of earthquakes. The Richter scalereading,

I

R, is given by the function R = log10

, where I is the intensity of the earthquakeand I0 is a

I0

low base-level intensity.

A Richter reading of 5.5 represents a strong earthquake capable of causing damage to

buildings.

(a) What is the Richter scale reading of an earthquake that is 10 times as strong?

(b) What is the Richter scale reading of an earthquake that is 100 times as strong?

(c) What is the Richter scale reading of an earthquake that is 1000 times as strong?

(d) An earthquake has a Richter scale reading of 7.9. How many times stronger is

thisearthquake than the one for which the reading is 5.5?

PROBLEM 2Resale Value(3 + 3 + 4 + 12 = 22 marks)

Cars lose value or depreciate from the moment that they are delivered to their first owner. In

the

early years they lose large amounts. As the price drops, the amount of depreciation reduces

also.

Adele has just bought a new car and is keen to investigate how well it will retain its value.

She

used an Internet database of car prices to obtain the median price of cars of her make and

model

of different ages. The results are shown in the table.

(a)Plot the data and comment on the result.

(b)Adele suspects that her make of car loses between 8 and 14% of the remaining value

eachyear. She initially uses a trial-and-error approach to find the depreciation rate, using

anexponential model.

Complete the following table to determine the rule for the value of the car, at time t years,for

depreciation (negative growth) rates of 8 to 14%.

(c)On the same set of axes as the plot for part a above, graph the remaining value, V,

fordepreciation rates of 8%, 10%, 12% and 14%. Which of these depreciation rates best

fitsthe data?

(d)An exponential model can have a rule of the form V(t) = a × bt.

(i)Show that, by taking the logarithm (of any base, k) of both sides, the result is

logkV(t) = logk(a) + tlogk(b).

(ii)Explain why, for an exponential model, the plot of logkV(t) versus t will result in

astraight line of gradient equal to logk(b) and vertical intercept equal to logk(a).

(iii)Hence plot logeV(t) versus t. Comment on the result. (Note: loge is entered as

LN formost CAS calculators.)

(iv)Use least squares regression to find the equation of the line of best fit.

(v)Hence find the values of a and b that best fit the data, and write the rule for V(t).

(vi)Compare the rules and depreciation rates found by using regression with those

foundusing trial and error.

PROBLEM 3

Drug doses

(4 + 6 + 6 + 6 = 22

marks)

A doctor prescribes a particular medication for a patient. Over a period of time the

medication is broken down and absorbed by the body. The level of medication, y, in the

patients blood at time t hours is given by y = A × 0.9t, where A mg/L is the initial dose.

(a)

Plot graphs of y against t for A = 0.1, 0.2, 0.3, 0.4. (Use t = 0, 1, 2, 3
8.) In each

case estimate the length of time it takes for the level of medication in the patients

blood to be halved. Does this value depend on A? Why?

(b)

Suppose that the dosage is such that the initial dose is A = 0.4 mg/L and this dosage is

repeated every 4 hours for 24 hours.

(i) Initiallyf1 = 0.4 ? 0.9t (the model for the first four hours, 0Statistics>Stat Claculations>Sinusoidal Regression.

CassioClassPad: Calc>Regression>Sinusoidal Reg

(c)

Estimate the height of the water at each of the following times:

(i) 1:30 pm

(d)

(e)

(ii)

3:30 pm

(iii)

11:30 pm

At what times in the 24 hour period is the water 4 metres in depth?

For what periods of time is the water depth increasing?

Question 4: Bungee Jump Practical Application Task( 5 marks)

In Unit 1, Week 11, students were asked to complete a Data Logging activity as part of

the graded task. This activity used the program called Logger Pro. The programming

licence allows current DECV students to install the program on their home computers.

It can be accessed from the Toolbox on the online site. In Unit 2, once again you are

expected to use Logger Pro to complete anapplication task. For this practical

application task involving a Bungee Jump, you will examine videos of real motion that

can be modelled using trigonometric equations.

Using the Logger Pro software programand the files from the Bungee Jump folder on

your course CD (under Further Resources), your task is to analysethe cyclical motion of

a spring. You are asked to model this motion using Logger Proand write a report. The

report need not be longer than two A4 sheets of paper. Further details are given after the

sample screen shown, and are also included in the Bungee Jump folder on your course

CD (under Further Resources). It is also available on the online course.

You may choose to extend the task by creating and analysing your own video of

cyclical motion such as a pendulum.

Once you learn to use the tools available with Logger Pro, the task is relatively straight

forward. However if you did not complete the Data Logging Activity in Unit 1, please

contact your teacher who will assist you in the use of the Logger Pro program.

You should find that modelling this real data will give you a much deeper understanding of

trigonometry than answering conventional textbook questions.

A sample screen from the Bungee Jumping activity usingLogger Pro is shown below.

The picture shows a frame from the video of a bungee jump. The force/time,

distance/time and velocity/time graphs of the motion are displayed on the right.

Bungee JumpData Logging with Logger Pro

To complete this activity you need to use the Logger Pro program as well as the Bungee

Jump series of files, including video files and data files, which are included in the CD you

received at the start of this unit. Both the Logger Pro program and the Bungee Jump files

are available on the online course.

This is what you need to do.

1. You should have Logger Pro already installed on your computer. If not, please install

it now. You will be prompted to install Quick Time as well. If you do not already

have Quick Time on your computer, you will need to agree to this.

2. Open the Bungee Jump files. It contains data gathered from a few experiments. It

also contains videos of these experiments and explanations about how to use the

Logger Pro program.

3. Open and play the three ~Using Logger Pro and the Introduction to Bungee files.

4. Open and view the Bungee 1.avi file to see the first bungee jump. Then open Bungee

1.cmbl file to see the results from this first Bungee jump using the Data Logger

program.

Finally open the Bungee 1 trial.cmbl file to see the final results and analysis of this

first bungee jump.

5. Choose Bungee 2 and model the force, distance and velocity using sine and cosine

curves using a similar process as was done for the first bungee jump.

6. Copy and paste photos, graphs and tables into a Word document and write up a report.

Include calculations stating the amplitude and period of the graphs, as well as the

equations for the distance/time, force/time and velocity/time graphs. State a few

distance, force and velocity values for specific times. The report should include an

introduction explaining the experiment, a main part with the explanation of the

models you found and a conclusion stating what you learned.

Please feel free to contact your teacher for any further assistance.

…

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