Mathematics AI SL
IB Questions
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Question 1 of 10
The intensity of volcanic eruptions is measured on the Volcanic Explosivity Index (VEI), with values typically between 0 and 8 where 8 is the most severe.
The Miller-Smith equation gives the average number of volcanic eruptions per year, \(N\), which have an index of at least \(M\). For a particular region the equation is
\(\log_{10} N = b - M\), for some \(b \in \mathbb{R}\)
This region has an average of 50 volcanic eruptions per year with an index of at least 2.
Find the value of \(b\).
The equation for this region can also be written as \(N = \frac{k}{10^M}\).
Find the value of \(k\).
Given \(0 < M < 8\), find the range for \(N\).
The expected length of time, in years, between volcanic eruptions with an index of at least \(M\) is \(\frac{1}{N}\).
Within this region the most severe volcanic eruption recorded had an index of 6.8.
Calculate the expected length of time between this eruption and the next eruption of at least this magnitude. Give your answer to the nearest year.
Question 2 of 10
Maria has a cat and buys cat food in large bags. She feeds her cat the same amount each day, and the amount of cat food remaining in the bag at the end of each day follows an arithmetic sequence.
After opening a new bag and feeding her cat, there were 85.5 cups of cat food remaining at the end of the fourth day, and 80 cups remaining at the end of the ninth day.
Find the number of cups of cat food fed to the cat per day.
Find the number of cups of cat food remaining in the bag at the end of the first day.
Calculate the number of days that Maria can feed her cat with one bag of food.
In 2021, Maria spent \(\$750\) on cat food. She expects that the amount she spends on cat food will increase at an annual rate of \(5.8\%\).
Determine the amount that Maria expects to spend on cat food in 2025. Round your answer to the nearest dollar.
Calculate the value of \(\sum_{n=1}^{10}\left(750 \times 1.058^{(n-1)}\right)\).
Describe what the value in part (d)(i) represents in this context.
Comment on the appropriateness of modelling this scenario with a geometric sequence.
Question 3 of 10
Environmental scientists are monitoring methane emissions from a landfill site. Let \(E\) be the emission rate in kilograms per hour, and \(t\) be the time in hours since monitoring began.
When \(E\) is plotted against \(t\), the total amount of methane produced is represented by the area between the graph and the horizontal \(t\)-axis.
The rate, \(E\), is measured over three hours. The results are shown in the following table:
| \text{boldsymbol t} | 0 | 0.6 | 1.2 | 1.8 | 2.4 | 3.0 |
|---|---|---|---|---|---|---|
| \text{boldsymbol E} | 25 | 40 | 55 | 35 | 20 | 45 |
Methane emission rates over time
Use the trapezoidal rule with an interval width of 0.6 to estimate the total amount of methane emitted during these three hours.
The actual amount of methane emitted during these three hours was 98 kilograms.
Calculate the percentage error of the estimate found in part (a).
Question 4 of 10
Maria has \(\$750\) in her savings account. She considers investing the money for 4 years with a bank. The bank offers an annual interest rate of \(1.8\%\) compounded monthly.
Calculate the amount of money Maria would have at the end of 4 years with the bank. Give your answer correct to two decimal places.
Instead of investing the money, Maria decides to buy a laptop that costs \(\$750\). At the end of 4 years, the laptop will have a value of \(\$45\). It may be assumed that the depreciation rate per year is constant.
Calculate the annual depreciation rate of the laptop.
Question 5 of 10
A city starts a tree-planting initiative where they plant 120 trees in the first month. In each subsequent month, they plant 25 more trees than the previous month.
| Month | Trees planted |
|---|---|
| 1 | 120 |
| 2 | 145 |
| 3 | 170 |
Number of trees planted in first three months
Calculate the number of trees to be planted in the 12th month.
Find the total number of trees to be planted in the first 12 months.
Calculate the mean number of trees planted per month during the first 12 months.
Question 6 of 10
In this question, give all answers correct to \(\mathbf{2}\) decimal places.
Marcus and Maria are planning to purchase a new apartment and need a loan of $225,000 from a bank. The loan term is 25 years with an annual interest rate of \(4.2\%\), compounded monthly. They will make fixed monthly payments at the end of each month.
Calculate their required monthly payment amount.
Find the remaining loan balance after the first 8 years.
Using your answers to parts (a) and (b)(i), calculate the total interest paid during the first 8 years.