IB MATH AI SL, Paper 1, May, 2023, TZ2, Solved Past Paper

Zaha is working on a project to design a bridge that will span across a river. She has made an estimation regarding the weight of the steel required for this bridge, which she believes to be approximately 53,632,000 kg. However, the actual weight of the steel needed for the bridge is known to be 55,625,000 kg. This discrepancy between the estimated and actual weight is the focus of the following questions.

In part (a), you are asked to calculate the percentage error in Zaha's approximation. Percentage error is a measure of how inaccurate a measurement is, standardized to how large the measurement is. It is calculated using the formula:

$\text{Percentage Error}=\left|\frac{\text{Estimated Value}-\text{Actual Value}}{\text{Actual Value}}\right|\times 100\mathrm{\%}$

In part (b), Zaha's design is used to construct five identical bridges. You are required to determine the total weight of the steel needed for these five bridges, rounded to three significant figures. Furthermore, you must express this total weight in scientific notation, which is a way of expressing numbers that are too large or too small to be conveniently written in decimal form.

Angel has a savings account with an initial amount of $520. He is considering investing this money for a period of 5 years with a bank that offers an annual interest rate of 1.2%, compounded quarterly. This means that the interest is calculated and added to the account balance four times a year, which affects the total amount of money accumulated over the investment period.

In part (a) of this question, you are required to calculate the total amount of money Angel would have in his account at the end of the 5-year period, taking into account the compound interest. The answer should be rounded to two decimal places.

In part (b), instead of investing the money, Angel decides to purchase a phone for $520. After 5 years, the phone's value is expected to depreciate to $30. You need to calculate the annual depreciation rate, assuming the rate is constant over the years. This involves determining how much the phone's value decreases each year as a percentage of its previous year's value.