How to Change the Subject of a Formula and Calculate Volumes

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Kimberley

26/02/2023

Maths

Block 2 - National 5 Maths

Subjects

Maths

Algebra

Straight line graphs

How to Change the Subject of a Formula and Calculate Volumes

Name: Knowunity
Brand: Knowunity

This guide covers key mathematical concepts including changing the subject of formulas, calculating volumes, interquartile range, standard deviation, straight line equations, and arc length and sector area calculations. It provides step-by-step examples to illustrate these important topics for students.

• Learn how to change the subject of a formula by applying opposite operations
• Practice calculating volume of a cylinder to significant figures using geometric formulas
• Understand how to calculate interquartile range of temperatures and interpret the results
• Master techniques for finding standard deviation, gradients, and equations of lines
• Apply formulas for arc length and sector area in practical examples

...

26/02/2023

355

block
Changing the Subject
1. Change the Subject of the formula
F = + + 4b₂
C
t+4b
F
t+46 CF
2
t = CF-4b
v = 2 p
to b
Opposite side
-> Oppos

View

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:

Calculate the volume of the cylinder $V₁$
Calculate the volume of the hemisphere $V₂$
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.

View

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:

Arrange the data in ascending order
Find the median $Q2$
Calculate Q1 and Q3
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.

View

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:

Calculate the mean of the data set
Find the difference between each data point and the mean
Square these differences
Sum the squared differences
Divide by $n-1$ , where n is the number of data points
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.

View

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:

Rearrange an equation into y = mx + c form
Identify the gradient and y-intercept from this form
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.

View

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:

Calculate the gradient using two given points
Use the point-slope form to write the equation
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.

View

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.

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Subjects

Maths

Algebra

Straight line graphs

Maths

•

355

•

26 Feb 2023

•

7 pages

How to Change the Subject of a Formula and Calculate Volumes

Kimberley

@kimberley_ncyf

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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:

Calculate the volume of the cylinder $V₁$
Calculate the volume of the hemisphere $V₂$
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.

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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:

Arrange the data in ascending order
Find the median $Q2$
Calculate Q1 and Q3
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.

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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:

Calculate the mean of the data set
Find the difference between each data point and the mean
Square these differences
Sum the squared differences
Divide by $n-1$ , where n is the number of data points
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.

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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:

Rearrange an equation into y = mx + c form
Identify the gradient and y-intercept from this form
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.

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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:

Calculate the gradient using two given points
Use the point-slope form to write the equation
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.

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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.

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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.

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How to Change the Subject of a Formula and Calculate Volumes

Volume Calculations

Interquartile Range

Standard Deviation

Straight Line Equations

Equation of a Line

Area of a Sector

Similar content

Can't find what you're looking for? Explore other subjects.

Knowunity is the #1 education app in five European countries

Knowunity has been named a featured story on Apple and has regularly topped the app store charts in the education category in Germany, Italy, Poland, Switzerland, and the United Kingdom. Join Knowunity today and help millions of students around the world.

4.9+

21 M

#1

950 K+

Still not convinced? See what other students are saying...

I love this app so much, I also use it daily. I recommend Knowunity to everyone!!! I went from a D to an A with it :D

The app is very simple and well designed. So far I have always found everything I was looking for :D

I love this app ❤️ I actually use it every time I study.

How to Change the Subject of a Formula and Calculate Volumes

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Volume Calculations

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Interquartile Range

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Standard Deviation

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Straight Line Equations

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Equation of a Line

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Area of a Sector

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Changing the Subject of a Formula

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