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Applications of Logarithms — Free MYP4 Mathematics (Extended) Practice Questions
1QuestionRichter scale (earthquakes)Assessment Practice
8 marks~12 minCriterion A
The table below shows the magnitude and amplitude ratio (relative to a reference earthquake of magnitude 0) for four earthquakes recorded at a seismology station.
Magnitude : 1.0, 2.0, 3.0, 4.0
Amplitude ratio : 10, 100, 1000, 10 000
The relationship between and follows the model .
Magnitude : 1.0, 2.0, 3.0, 4.0
Amplitude ratio : 10, 100, 1000, 10 000
The relationship between and follows the model .
a
Determine the value of when , giving your answer to 3 significant figures. [3]
b
A second station records an earthquake with amplitude ratio . A third station, at a different location, records the same earthquake and measures an amplitude ratio 4 times greater. Find the magnitude recorded by the third station, giving your answer to 3 significant figures. [3]
c
A seismologist claims that an earthquake of magnitude 6.0 has an amplitude ratio exactly 1000 times greater than one of magnitude 3.0. Justify whether this claim is correct. [2]
Solutions
2QuestionSolving real-life exponential problemsAssessment Practice
8 marks~12 minCriterion D
A patient receives a 200 mg dose of a medication that decays exponentially. The amount (in mg) remaining after hours is modelled by
The diagram shows the graph of for , together with three observed measurements from the patient's blood tests.
The doctor requires the drug concentration to fall below 50 mg before administering the next dose.
The diagram shows the graph of for , together with three observed measurements from the patient's blood tests.
| (hours) | 2 | 6 | 10 |
|---|---|---|---|
| Predicted (mg) | 144.5 | 75.4 | 39.4 |
| Observed (mg) | 138 | 80 | 46 |
The doctor requires the drug concentration to fall below 50 mg before administering the next dose.
a
Show that the predicted time at which the drug concentration first falls below 50 mg is hours. [3]
b
Find the percentage error between the predicted and observed concentrations at hours, giving your answer to one decimal place. Hence comment on whether the model's prediction at hours is reliable enough to use as the basis for the dosing decision, referring to the errors in your table. [3]
c
State one assumption of this exponential decay model. Advise the doctor whether the predicted dosing time of hours should be used to make the dosing decision, justifying your answer by referring to the model's limitations outside . [2]
Solutions
3QuestionSolving real-life exponential problemsAssessment Practice
9 marks~14 minCriterion C
A sample of 100 grams of a radioactive isotope decays by half every 8 days. The table below shows the mass remaining after each half-life period.
| Number of half-lives () | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Mass remaining (g) | 100 | 50 | 25 | 12.5 | 6.25 |
a
Find the mass remaining after 7 half-lives. Give your answer in grams. [2]
b
The mass grams after half-lives can be written in the form
Show that , and hence find the number of complete half-lives required for the mass to fall below 1 gram for the first time. [4]
Show that , and hence find the number of complete half-lives required for the mass to fall below 1 gram for the first time. [4]
c
A student claims that after 20 half-lives the mass remaining is exactly 0 grams, because "the sample keeps halving and must eventually reach zero." Justify whether this claim is mathematically correct, referring to the structure of the exponential model. [3]
Solutions
4QuestionGrowth and decay problemsAssessment Practice
10 marks~15 minCriterion B
The diagram shows the first four figures in a sequence built by adding a new border of squares around the previous figure.
| Figure | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Total squares | 1 | 5 | 13 | 25 |
a
Complete the table below by finding the number of squares added when going from Figure to Figure , for . Describe the pattern in these differences. [3]
Squares added: \_\_\_, \_\_\_, \_\_\_
Squares added: \_\_\_, \_\_\_, \_\_\_
b
Show that the total number of squares in Figure is given by
[4]
[4]
c
A student claims that Figure 20 will have more than 750 squares. Justify whether this claim is correct, and find the smallest value of for which . [3]
Solutions
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