Volume Calculations

This page focuses on volume calculations for various 3D shapes, providing essential formulas and a practical example.

Vocabulary: Volume is the amount of three-dimensional space occupied by an object.

The page lists key volume formulas:

• Prism: V = Ah (Area of base × height)
• Cylinder: V = πr²h (π × radius² × height)
• Cuboid: V = lbh (length × breadth × height)

Example: A complex shape consisting of a cylinder and a hemisphere is presented. The problem requires calculating the total volume to 3 significant figures.

The solution is broken down into steps:

1. Calculate the volume of the cylinder (V₁)
2. Calculate the volume of the hemisphere (V₂)
3. Sum the two volumes and round to 3 significant figures

Highlight: When solving complex volume problems, break the shape into simpler components and calculate their volumes separately before combining.

Interquartile Range

This page explains the concept of interquartile range (IQR) and its application in comparing data sets.

Definition: The interquartile range is a measure of statistical dispersion, calculated as the difference between the upper (Q3) and lower (Q1) quartiles.

The formula for IQR is presented: IQR = Q3 - Q1

Example: A set of midday temperatures in Glasgow is given: 3, 3, 3, 4, 4, 5, 6, 7, 9, 10. The problem requires calculating the median and IQR.

The solution demonstrates how to:

1. Arrange the data in ascending order
2. Find the median (Q2)
3. Calculate Q1 and Q3
4. Compute the IQR

Highlight: The IQR is useful for comparing the spread of data between different sets, as shown in the comparison between Glasgow and Edinburgh temperatures.

Standard Deviation

This page covers the calculation of mean and standard deviation for a given data set.

Definition: Standard deviation is a measure of the amount of variation or dispersion of a set of values.

The page provides a step-by-step guide to calculating standard deviation:

1. Calculate the mean of the data set
2. Find the difference between each data point and the mean
3. Square these differences
4. Sum the squared differences
5. Divide by $n-1$, where n is the number of data points
6. Take the square root of the result

Example: A data set of 14, 17, 15, 23, 20, 19 is used to demonstrate the calculation process.

Highlight: The sum of the differences between each data point and the mean should always equal zero, serving as a check for calculations.

Straight Line Equations

This page focuses on the equation of a straight line and how to determine its gradient and y-intercept.

Vocabulary: The general form of a straight line equation is y = mx + c, where m is the gradient and c is the y-intercept.

The page demonstrates how to:

1. Rearrange an equation into y = mx + c form
2. Identify the gradient and y-intercept from this form
3. Calculate the gradient using two points on the line

Example: The equation 2x - 5y = 20 is rearranged to find the gradient and y-intercept.

Highlight: The gradient formula m = $y₂ - y₁$ / $x₂ - x₁$ is crucial for finding the slope between two points.

Equation of a Line

This page extends the concept of straight lines, focusing on finding the equation of a line passing through two points.

Vocabulary: The point-slope form of a line equation is y - y₁ = m$x - x₁$, where (x₁, y₁) is a point on the line and m is the gradient.

The page demonstrates the process of finding a line equation:

1. Calculate the gradient using two given points
2. Use the point-slope form to write the equation
3. Simplify the equation to y = mx + c form

Example: Find the equation of the line passing through $-1, -7$ and (4, 3).

The page also introduces arc length calculations:

• Formula: l = θ/360° × πd, where θ is the angle in degrees and d is the diameter

Highlight: The arc length formula can be rearranged to find unknown angles or diameters when other variables are given.

Area of a Sector

This final page covers the calculation of sector area in a circle.

Definition: A sector is a part of a circular disk enclosed by two radii and an arc.

The formula for the area of a sector is presented: A = θ/360° × πr², where θ is the angle in degrees and r is the radius.

Example: Calculate the area of a sector with radius 3cm and angle 150°.

The page also shows how to:

• Find an unknown angle given the area and radius
• Calculate the radius when given the area and angle

Highlight: Sector area problems often involve rearranging the formula to solve for different variables, similar to changing the subject of a formula.

Changing the Subject of a Formula

This page introduces the concept of changing the subject of a formula in mathematics. It demonstrates the process using two examples.

Definition: Changing the subject of a formula involves rearranging an equation to isolate a different variable as the subject.

The first example shows how to change the subject of the formula F = C/$t+4b$ + 4b² to t. The process involves applying opposite operations to isolate t.

Example: To change F = C/$t+4b$ + 4b² to make t the subject, subtract 4b² from both sides, multiply by $t+4b$, and then subtract 4b to isolate t.

The second example demonstrates changing the subject of P = my²/2 to v. This involves multiplying both sides by 2, dividing by m, and then taking the square root.

Highlight: When changing the subject of a formula, always perform the same operation on both sides of the equation to maintain equality.