# AI SL: Paper 1 (TZ2 May 2021)

TLDRThis video script offers a comprehensive educational session, covering a variety of mathematical problems and concepts. It begins with a detailed explanation of drug dosage calculation and the percentage of drug elimination from the body over time. The script then delves into geometric problems involving the distance between points, coordinates of midpoints, and the height of 3D figures. It continues with surface area calculations for a storage container and a box with a half-cylinder lid, followed by trigonometric problems related to angles of depression and elevation. The session also includes a statistical analysis of sick days using a box and whisker plot, a t-test for comparing the weights of chinchilla and sable rabbits, and a compound interest problem. The script concludes with a goodness of fit test for a newspaper vendor's sales model, a quadratic function exploration, and a calculus problem on profit maximization in electric car production.

### Takeaways

- π The initial dose of a drug can be found by evaluating the drug concentration function at the earliest time point, which is when T equals zero.
- π§ To determine the percentage of a drug that leaves the body each hour, it's important to understand the function's mechanics and not just take the number adjacent to the initial dose at face value.
- π For calculating the amount of drug remaining in the body after a certain number of hours, plug in the specific time (T) into the drug concentration function and evaluate.
- π The distance between two points in 3D space can be calculated using the distance formula, ensuring to correctly identify corresponding X, Y, and Z coordinates.
- π Finding the coordinates of a midpoint, such as a station between two others, involves averaging the respective coordinates of the two points.
- π To find the height of a point above the ground in a 3D scenario, identify the vertical coordinate (often Z) and apply the correct units.
- π¨ When calculating the area to be painted for a storage container with a half-cylindrical lid, consider the exterior surface area, including the curved surface of the cylinder and the circles at the ends.
- π The angle of depression from a point above to a point on the ground is equal to the angle of elevation from the ground point to the point above, crucial for understanding relative positions.
- π To find the distance between two points when one has moved from an initial to a final position, use the sine rule or trigonometric principles to relate the angles and sides of the triangles formed.
- π’ In a box and whisker diagram, the minimum, lower quartile (Q1), median (Q2), upper quartile (Q3), and maximum are represented by specific points on the graph, providing a quick summary of data distribution.
- π The percentage of employees who took fewer or more sick days can be inferred from the spread of the box and whisker diagram, but it's important to correctly interpret the percentages represented by each section.

### Q & A

### What is the initial dose of the drug mentioned in the script, and how is it calculated?

-The initial dose of the drug is 23 milligrams. It is calculated by plugging in T equals 0 into the given function, which simplifies to 23 times 1, hence the initial dose is 23 milligrams.

### What is the percentage of the drug that leaves the body each hour, and why is the common interpretation incorrect?

-The common interpretation that 85 percent of the drug leaves the body each hour is incorrect. The correct interpretation is that 15 percent of the drug remains in the body each hour, which means 85 percent is incorrectly associated with the loss, rather than the remaining amount.

### How is the amount of drug remaining in the body after 10 hours calculated, and what is the result?

-The amount of drug remaining after 10 hours is calculated by plugging in T equals 10 into the function. The result is 4.52 milligrams, which is obtained by multiplying 23 by 0.85 to the power of 10.

### What is the formula used to calculate the distance between two points in a 3D space, and how is it applied in the script?

-The formula used to calculate the distance between two points in a 3D space is the 3D distance formula: β((x2 - x1)Β² + (y2 - y1)Β² + (z2 - z1)Β²). In the script, it is applied by identifying the coordinates of points A and B and plugging the values into the formula to find the distance between them.

### How is the height of station M above the ground determined, and what is its value?

-The height of station M above the ground is determined by identifying the z-coordinate of the midpoint between stations A and B. Since the z-axis represents the vertical direction, the height of station M is the z-coordinate of point M, which is 125 meters.

### What is the surface area of a half-cylinder, and how is it calculated in the script?

-The surface area of a half-cylinder includes the curved surface area and the area of the two half-circles at the ends. In the script, the curved surface area is calculated as Οrh, and the area of the two half-circles is calculated as ΟrΒ², with both results being halved because it's a half-cylinder.

### What is the angle of depression from point H to point C in the helicopter scenario, and how is it found?

-The angle of depression from point H to point C is 25 degrees. It is found by recognizing that the angle of depression is the same as the angle of elevation when observing point H from point C, which is given as 25 degrees in the script.

### How is the distance from point A to point C calculated in the helicopter scenario?

-The distance from point A to point C is calculated using the sine rule, which relates the lengths of sides of a triangle to the sines of its opposite angles. The calculation involves finding the unknown side (AC) using the known angle (25 degrees) and the opposite side (380 meters).

### What is the process to determine the location of the new bus stop on the road between two schools, and where should it be located?

-The process involves finding the perpendicular bisector of the line segment between the two schools. The bus stop should be located at the point where this perpendicular bisector intersects the given road, ensuring equal distances from both schools.

### How is the equation of the perpendicular bisector found, and what is its final form?

-The equation of the perpendicular bisector is found by first determining the slope of the line segment between the two schools and then finding the negative reciprocal of this slope. Using the midpoint of the segment and the new slope, the point-slope form of the line equation is used to derive the final form of the perpendicular bisector, which is y = -3x + 46.

### Outlines

### π€ Mathematical Drug Dosage Analysis

The paragraph discusses the mathematical modeling of a drug's concentration in the body over time after injection. It explains the concept of initial dose calculation, the percentage of drug remaining after a certain period, and the importance of understanding the function's behavior. The speaker clarifies misconceptions about the percentage of drug elimination per hour and demonstrates how to calculate the amount of drug left in the body after 10 hours using the formula provided.

### π Calculating Distance and Coordinates in 3D Space

This section of the script focuses on using mathematical formulas to find distances between points and calculate coordinates in three-dimensional space. It introduces the distance formula and midpoint formula, applying them to find the distance between two railway stations and the coordinates of a midpoint for a new station. The explanation emphasizes the importance of correctly pairing coordinates and understanding the context of 3D space.

### π¨ Surface Area Calculation for a Storage Container

The script explains how to calculate the total exterior surface area of a storage container that consists of a box with a half-cylinder lid. It details the process of identifying which surfaces need to be calculated and which do not, such as the bottom of the box. The explanation includes the formulas for the surface area of a rectangle and a cylinder, and it demonstrates the calculation of the curved surface area of the half-cylinder and the area of the circular ends.

### π Angles and Distances Involving a Helicopter and Swimmers

This paragraph explores geometric relationships involving a helicopter hovering above a lake and a swimmer observing the helicopter from different points. It discusses the concept of angle of depression and elevation, using them to determine the angle of the helicopter from the swimmer's perspective. The script then delves into the use of sine rule to find the distance from the lake's surface to the swimmer's point and the distance between two points on the shore, employing trigonometric principles to solve for unknown distances.

### π Interpreting a Box and Whisker Diagram for Sick Days

The script provides an analysis of a box and whisker diagram representing the number of sick days taken by employees in a company over a year. It explains how to read the diagram to find the minimum, lower quartile, median, and upper quartile values. The explanation also addresses a claim about the percentage of employees who took fewer or more sick days than certain values, using the diagram's proportions to refute the claim.

### π Locating a Bus Stop on a Road Using Perpendicular Bisector

The paragraph describes a problem involving finding the optimal location for a bus stop equidistant from two schools represented by points on a graph. It explains the concept of a perpendicular bisector and how to find it using the slope of the line connecting the two schools and the midpoint formula. The explanation includes the process of deriving the equation of the perpendicular bisector and finding its intersection with the given road to determine the bus stop's location.

### π Determining the Range of a Function and Evaluating Its Inverse

This section discusses the process of finding the range of a given function by analyzing its graph and identifying the possible y-values it can take. It also covers the concept of an inverse function and demonstrates how to derive it by swapping and solving for the variable. The explanation includes an example of evaluating the inverse function at a specific point to find its value.

### π° Statistical Analysis of Rabbit Weights Using a T-test

The script outlines a statistical test to determine if there is a significant difference in the weights of two types of rabbits, using a t-test. It explains how to formulate the null and alternative hypotheses, perform the test using a calculator, and interpret the p-value obtained from the test. The explanation includes the steps to conclude the test based on the comparison of the p-value with the significance level.

### π³ Calculating the Area and Length of a Sector in a Garden

This paragraph explains how to calculate the area of a lawn shaped like a sector of a circle and the length of the arc that forms its boundary. It introduces the formulas for the area of a sector and the length of an arc, and demonstrates their application using the given angle and radius. The explanation includes the process of plugging the values into the formulas and performing the calculations to find the area of the lawn and the length of the footpath around it.

### πΆ Compound Interest Calculation for Savings Accounts

The script discusses the concept of compound interest, specifically when applied to savings accounts. It explains how to calculate the future value of an investment using the compound interest formula, taking into account the principal amount, interest rate, compounding frequency, and the number of years. The explanation includes an example of calculating the amount in a college savings account after five years with a given interest rate compounded half-yearly.

### π Goodness of Fit Test for Newspaper Sales Prediction

This paragraph describes a goodness of fit test used to evaluate a newspaper vendor's model for predicting daily sales. It explains the process of estimating expected sales per day, calculating the chi-squared value, and comparing it to a critical value to determine if the model is suitable. The explanation includes the steps for setting up the test, interpreting the results, and concluding whether the model fits the actual sales data.

### π Solving a Quadratic Function for a Parabola's Properties

The script explains how to solve for the coefficients of a quadratic function representing a parabola, given its vertex and the points where it intersects the x-axis. It demonstrates the process of setting up a system of equations based on these points, solving the system to find the values of the coefficients, and using the formula for the axis of symmetry to find the line that runs through the vertex of the parabola.

### π Calculating Profit Function for Electric Car Production

This paragraph discusses the process of deriving a profit function for a company that produces electric cars, based on the rate of change of profit with respect to the number of cars produced. It explains the concept of integration to find the original profit function from its derivative, using given values to solve for the constant of integration. The explanation includes an example of how to determine the profit function and use it to analyze the company's profit at different production levels.

### Mindmap

### Keywords

### π‘Medicinal drug

### π‘Initial dose

### π‘Percentage of drug leaving the body

### π‘Distance formula

### π‘Midpoint

### π‘Surface area

### π‘Angle of depression

### π‘Box and whisker diagram

### π‘Perpendicular bisector

### π‘Compound interest

### π‘Goodness of fit test

### π‘Quadratic function

### π‘Integral

### Highlights

The initial dose of a medicinal drug is calculated using the formula DT at T equals zero.

The percentage of drug leaving the body each hour is intuitively determined by understanding the function's impact on the initial dose.

Calculating the amount of drug remaining after a certain time involves careful attention to the function's variables and units.

Finding the distance between two points involves applying the distance formula correctly with given coordinates.

The coordinates of a midpoint between two stations are calculated using the midpoint formula in a 3D space.

Determining the height of a point in a 3D coordinate system requires understanding the orientation of axes.

The total exterior surface area of a storage container is calculated by considering all relevant sides and shapes.

The curved surface area of a half-cylinder is computed using the formula for a full cylinder and adjusting for the half shape.

The angle of depression from a point is equal to the angle of elevation to the same point, a key concept in trigonometry.

Finding the distance from one point to another involves using trigonometric principles and the sine rule effectively.

Calculating the speed of an object involves converting time units and applying the correct formula for rate.

Interpreting a box and whisker diagram requires understanding the representation of minimum, lower quartile, median, and maximum values.

The percentage of employees who took fewer or more sick days is inferred from the box and whisker diagram's distribution.

Finding the equation of a perpendicular bisector involves understanding the relationship between slopes of perpendicular lines.

Determining the location of a bus stop equidistant from two schools involves finding the intersection of a perpendicular bisector and a given line.

The range of a function is determined by analyzing the graph and identifying the minimum and maximum y-values.

Calculating the value of an inverse function involves manipulating the original function's equation to solve for the inverse.

Conducting a t-test involves setting up null and alternative hypotheses, calculating the test statistic, and comparing it to a critical value or p-value.

The length of an arc in a circle is calculated using the proportion of the circle's angle and the circle's radius.

The area of a sector and a triangle within a circle are used to determine the area of a shaded lawn in a garden.

Compound interest is calculated using the formula that includes principal amount, interest rate, compounding frequency, and time.

Investment growth is modeled by setting up an equation based on the expected future value and solving for the unknown interest rate.

Goodness of fit tests are used to evaluate a model's accuracy by comparing observed and expected values.

The symmetry of a parabola is utilized to find unknown points on the graph by recognizing equal distances from the vertex.

The values of a, b, and c in a quadratic function are determined by creating and solving a system of equations derived from given points.

The axis of symmetry for a parabola is calculated using the formula that relates the coefficients a and b.

Finding the profit function involves integrating the rate of change of profit with respect to the number of cars produced.

The change in profit with varying production levels is analyzed by examining the derivative of the profit function.