Learning activity 1.3: Working with formulas

Aerial photo of soccer stadium.

Introduction

Building sporting venues, designing electronics, and constructing buildings all involve the use of mathematical formulas. The power of math is amazing and there is no limit to what it allows us to do. In this learning activity, we will explore formulas together. Formulas allow numerous real-world problems to be studied. We will examine formulas and solve for unknown or “missing” values. There are many different formulas that allow you to solve for missing values. We will begin by examining some examples involving concepts that you have considered in earlier Math courses – area and volume.

If you need a quick refresher: perimeter is the distance around a shape (think fencing), area is the surface covered by a shape (think carpeting), and volume is the space occupied (think amount of water in a pool). Review these concepts by completing the following problems in your notebook. Compare your solution to the suggested answers.

Notebook

Determine the volume of a rectangular prism if the length is 6 metres, the width is 3.5 metres and the height is 2 metres. Recall that the formula for the volume of a rectangular prism is  $Volume = (length) \times (width) \times (height)$ or $V = l \times w \times h$ .

$V = l \times w \times h$

$V = ( 6 ) ( 3.5 ) ( 2 ) $

$V = 42$

The volume of the rectangular prism is $42 m^{3}$ .

Calculate the area of a triangle with a base length of 4.5 cm and a height of 6.4 cm. Recall that the formula for the area of a triangle is $Area = \frac{(base) \times (height)}{2}$ or $A = \frac{b \times h}{2}$ .

$A = \frac{b \times h}{2}$

$A = \frac{(4.5) (6.4)}{2}$

$A = \frac{28.8}{2}$

$A = 14.4$

The area of the triangle is $14.4 {cm}^{2}$ .

Photo of the hands of a person working on a laptop.

Did you know, mathematicians aren’t the only people who use formulas? Formulas are used by many individuals - architects, meteorologists, chefs, construction workers, electricians, carpenters, machinists, mechanics, financial analysists, and chemists - just to name a few.

Join the discussion

Use a search engine of your choice to find your favourite recipe and share it with your classmates. Explain how you would use your math knowledge from this course to make that recipe.

Now let’s find out more about formulas.

To be able to effectively work with formulas, you will need to use your algebraic skills. In this learning activity we will consider the two different approaches for working with formulas:

Substituting all known values first then solving for the unknown
Isolating the unknown first, then substituting all known values.

Let us begin by identifying some commonly used functions. In this learning activity, we will be studying linear, quadratic, and exponential functions.

What type of equation is it?

Linear equations:

are equations where the $x$ variable has an exponent of 1 which is not generally written since $x$ and $x^{1}$ are the same
in the form $y = m x + b$ where $m$ represents the slope of the line and $b$ represents the $y -$ intercept
graph as a straight line

Example: $y = 5 x - 1$

Quadratic equations:

are equations where the highest exponent of $x$ is 2
in the form $y = a x^{2} + b x + c$ ; $a \neq 0$
graph as a parabola

Example: $y = 5 x^{2} + 2 x - 1$

Exponential equations:

are equations that have the $x$ variable as an exponent
in the form $y = a b^{x}$
graph as a smooth curve that can open either up or down

Example: $y = 3 (2)^{x}$

Methods for solving

As mentioned previously, there are two common approaches to solve for an unknown variable. Let us consider both methods and then you can decide which works best for you. First, we will solve by substituting in all given values and in the second method, we will first rearrange the equation to isolate the unknown variable.

To illustrate this, let’s examine a formula that helps us understand money management in terms of loans and savings. We will revisit this application in Unit 4 when we learn about personal finance. The compound interest formula $A = P (1 + i)^{n}$ will be used in the following examples.

Compound Interest Formula, $A = P (1 + i)^{n}$

$A$ represents the total amount of money
$P$ represents the principal which is the amount that is initially borrowed or invested
$i$ represents the interest rate per compounding period and must be expressed in decimal form
$n$ represents the number of compounding periods

Image of a soccer ball resting on 500 Euro banknotes.

Substituting all given values first

Example 1

Using the formula $A = P (1 + i)^{n}$ , calculate the principal if $A = $ 2,500$ , $i = 0.06$ (which means the interest rate was 6%) and $n = 10$ .

The following example will present one step at a time in order to give you time to think about each step.

Substitute what is known.

We were given: $A = 2,500$ , $i = 0.06$ , $n = 10$

$2,500 = P (1 + 0.06)^{10}$

Simplify by adding the numbers in brackets

$2,500 = P (1.06)^{10}$

Divide both sides of the equation by $(1.06)^{10}$ to isolate the variable $P$

$\frac{2,500}{{(1.06)}^{10}} = \frac{P {(1.06)}^{10}}{{(1.06)}^{10}}$

$\frac{2,500}{{(1.06)}^{10}} = P$

Switching the sides around to get the variable on the left gives us:

$P = \frac{2,500}{{(1.06)}^{10}}$

$P = 1,395.986942$

(this is the calculator answer but since we are dealing with money the final answer should be rounded accurately to 2 decimal places and include units)

$P = 1,395.99$

Therefore the principal is $1,395.99

Example 2

Using the formula $A = P (1 + i)^{n}$ , calculate $n$ if $A = $ 20,000$ , $P = $ 5,000$ and $i = 0.07$ . Round your answer to two decimal places.

Solution:

Substitute what is known, in this example $A = 20,000$ ; $P = 5,000$ ; $i = 0.07$

$20,000 = 5,000 (1 + 0.07)^{n}$

Simplify.

$20,000 = 5,000 (1.07)^{n}$

This is an exponential equation since the variable ( $n$ in this case) is in the exponent so we will need to isolate the $(1.07)^{n}$ first.

Divide both sides by $5,000$ to isolate the $(1.07)^{n}$

$\frac{20,000}{5,000} = \frac{5,000 (1.07)^{n}}{5,000}$

$4 = (1.07)^{n}$

$(1.07)^{n} = 4$

(In this step, the left and right sides of the equation were interchanged so the variable was on the left side of the equation.)

Graph " $y = 1.0 7^{n}$ " and " $y = 4$ " using GeoGebra tool determine the point of intersection.

Alternatively, this could be solved using “guess and check” on your calculator.

The following grid has both $y = 1.0 7^{n}$ and $y = 4$ graphed.

The point of intersection is (20.49, 4). Ordered pairs are usually written $(x, y)$ but in this case, the variables in the equation are $n$ and $y$ so $n = 20.49$ .

Since $n = 20.49$ and there cannot be a decimal number of compounding periods we round up and obtain 21 compounding periods.

Example 3

Using the formula $A = P (1 + i)^{n}$ , calculate the value for $i$ if $A = $ 5,675$ , $P = $ 5,000$ and $n = 2$ . Round your answer to two decimal places.

Solution:

Substitute what is known, in this example $A = 5,675$ ; $P = 5,000$ ; $n = 2$

$5,675 = 5,000 (1 + i)^{2}$

Divide both sides of the equation by $5,000$ to isolate the variable that is in the brackets

\frac{5,675}{5,000} = \frac{5,000 (1 + i)^{2}}{5,000}

$1.135 = (1 + i)^{2}$

Left and right sides of the equation are interchanged here to have the variable, $i$ on the left side

$(1 + i)^{2} = 1.135$

Square root both sides to eliminate the exponent

$\sqrt{(1 + i)^{2}} = \sqrt{1.135}$

$1 + i = 1.065363788$

Subtract 1 from both sides to solve for $i$ and then multiply that value by $100$ to convert into percent form.

$1 + i = 1.065363788$

$i = 0.065363788$ (as a decimal)

$i = 6.5363788$ (as a percent)

$i = 6.54 %$ (as a percent rounded to 2 decimal places)

In this last example you needed to square root both sides of the equation in order to solve. What if the exponent is greater than 2? You will need to use your knowledge of rational exponents to solve the equation by first determining the reciprocal of the exponents involved. We will explore how this works in the following example.

Example

Step 1

Given $x^{4} = 20$ , solve for $x$ accurate to two decimal places.

Step 2

Determine the reciprocal of the exponent on $x$ . In this case the exponent is 4 so the reciprocal of 4 is $\frac{1}{4}$ .

Step 3

From Step 2 we know the reciprocal is $\frac{1}{4}$ so apply that to both sides of the equation to get

{(x^{4})}^{\frac{1}{4}} = {(20)}^{\frac{1}{4}}

$x = (20)^{\frac{1}{4}}$

$x = \sqrt[4]{20}$

Step 4

Use your calculator to find the 4^th root of 20 or you can also use the exponent key.

Step 5

Using either method from Step 4, you should obtain an answer of $±2.114742527$ and rounded to 2 decimal places you will get $x = ±2.11$ .

Notebook

You can use your notebook to complete each of the following questions. Compare your work with the suggested answers to check your understanding.

Given $x^{9} = 87$ , solve for $x$ accurate to two decimal places.

$x^{9} = 87$

$(x^{9})^{\frac{1}{9}} = (87)^{\frac{1}{9}}$

$x = (87)^{\frac{1}{9}}$

$x = \sqrt[9]{87}$

$x = 1.64$

Isolate then substitute

Image of a woman in running clothes stopping at the bottom of a flight of stairs to consult her devices which show her BPM is 120, time is 55:08 minutes, kCal burned is 508, number of steps is 10,000. — Press here for long description(Open in new window)

In some cases, it may be beneficial to isolate the desired variable first and then substitute.

Example

Rearranging the equation to isolate the unknown variable

Many everyday objects, like soup cans, are in the shape of a cylinder. Let’s explore different sizes of cylinders.

Four cylinders need to be produced, all with a height of 1.2 m, but with varying volumes. Determine the radius (to two decimal places) required to construct cylinders with the volumes given in this table.

	Cylinder A	Cylinder B	Cylinder C	Cylinder D
Required volume	$0.5 m^{3}$	$1.0 m^{3}$	$1.5 m^{3}$	$2.0 m^{3}$

If you substituted the known values first then solved for the unknown, you would have to do this four times. It is more efficient to solve for the unknown (radius, $r$ ) first and then substitute the known values.

The volume of a cylinder is given by the formula $V = π r^{2} h$ . In this example $h = 1.2 m$ can be substituted into the formula which leaves just one variable to isolate.

$V = π r^{2} h$

$V = π r^{2} (1.2)$

$V = 1.2 π r^{2}$

Divide both sides by $1.2 π$ to isolate $r^{2}$ .

$\frac{1.2 π r^{2}}{1.2 π} = \frac{V}{1.2 π}$

$r^{2} = \frac{V}{1.2 π}$

Take the square root of both sides.

$r^{2} = \frac{V}{1.2 π}$

$\sqrt{r^{2}} = \sqrt{\frac{V}{1.2 π}}$

$r = \sqrt{\frac{V}{1.2 π}}$

Now you can use the formula $r = \sqrt{\frac{V}{1.2 π}}$ to calculate the radius of the four cylinders.

Notebook

In your notebook, copy the following table. Use the new formula, $r = \sqrt{\frac{V}{1.2 π}}$ , to calculate the radius of each of the four cylinders. Check your answers by pressing the “Show Answers” bar.

	Cylinder A	Cylinder B	Cylinder C	Cylinder D
Required volume	$0.5 m^{3}$	$1.0 m^{3}$	$1.5 m^{3}$	$2.0 m^{3}$
Radius	$r = \sqrt{\frac{0.5}{1.2 π}}$ $r = 0.364 m$	$r = \sqrt{\frac{1}{1.2 π}}$ $r = 0.515 m$	$r = \sqrt{\frac{1.5}{1.2 π}}$ $r = 0.631 m$	$r = \sqrt{\frac{2}{1.2 π}}$ $r = 0.728 m$

Cylinder A

Cylinder B

Cylinder C

Cylinder D

Required volume

$0.5 m^{3}$

$1.0 m^{3}$

$1.5 m^{3}$

$2.0 m^{3}$

Radius

$r = \sqrt{\frac{0.5}{1.2 π}}$

$r = 0.364 m$

$r = \sqrt{\frac{1}{1.2 π}}$

$r = 0.515 m$

$r = \sqrt{\frac{1.5}{1.2 π}}$

$r = 0.631 m$

$r = \sqrt{\frac{2}{1.2 π}}$

$r = 0.728 m$

Solving multi-step problems in real-world applications

Two individuals playing chess with one being older than the other.

In mathematics, there can be more than one way to solve a problem. There is often an “easiest” way (although the easiest way could vary from person to person). Being able to manipulate formulas allows you to solve problems more efficiently. The following examples illustrate how to solve multi-step problems.

Example 1

A cylindrical tank is to be covered with two coats of paint including the top and bottom. The cylinder is 3.5 m high and has a radius of 2.4 m. If the formula for surface area of a cylinder is $S.A. = 2 π r (r + h)$ , determine:

a) the total surface area to be painted

b) the amount of paint needed, to the nearest litre if 1 L covers 18 m².

Determining the total surface area could be done in one of two ways. Which method do you prefer? Why?

Method 1	Method 2
For one coat: $S. A. = 2 π r (r + h)$ $= 2 π (2.4) (2.4 + 3.5)$ $= 2 π (2.4) (5.9)$ $= 88.97 m^{2}$ For two coats: $S A = 88.97 \times 2$ $= 177.94 m^{2}$	For two coats, the adjusted formula would be: $S. A. = 2 [2 π r (r + h)]$ $= 2 [2 π (2.4) (2.4 + 3.5)]$ $= 2 (88.97)$ $= 177.94 m^{2}$

Method 1

Method 2

For one coat:

$S. A. = 2 π r (r + h)$

$= 2 π (2.4) (2.4 + 3.5)$

$= 2 π (2.4) (5.9)$

$= 88.97 m^{2}$

For two coats:

$S A = 88.97 \times 2$

$= 177.94 m^{2}$

For two coats, the adjusted formula would be:

$S. A. = 2 [2 π r (r + h)]$

$= 2 [2 π (2.4) (2.4 + 3.5)]$

$= 2 (88.97)$

$= 177.94 m^{2}$

The amount of paint required, in litres, can be calculated by dividing the total surface area by $18 m^{3}$

Paint required $= 177.94 \div 18$

$= 9.89$

Therefore the amount of paint required is 10 L.

Example 2

The following cylindrical culvert tube is to be constructed of concrete 0.25 m thick.

Diagram is a long hollow cylinder, like a metal pipe, with walls of a certain thickness. The length of the cylinder is labelled h. The radius of the inner cylinder is labelled rinner and a line segment is drawn from the centre of the circle at one end of the cylinder to the inside wall of the cylinder. The radius of the outer cylinder is labelled rtotal and a line segment is drawn from the centre of the circle at the other end of the cylinder to the outside wall of the cylinder. — Press here for long description(Open in new window)

If the length of the tube is 12 m and the inner radius is 2.5 m, determine the volume of concrete needed. The formula for the volume of a cylinder is $V = π r^{2} h$ .

To solve this problem, think of the concrete tube as two cylinders: an outer “total volume” cylinder and an empty “inner volume” cylinder. To get the volume of the concrete, you must subtract the empty cylinder from the total volume.

$V_{c o n c r e t e} = V_{t o t a l} - V_{i n n e r}$

To use the formula for the volume of a cylinder, you need to know the height and the radius of both cylinders.

Height of both cylinders is given as 12 m so $h = 12$

Radius of inner cylinder is given as 2.5 m so $r_{i n n e r} = 2.5$

Radius of total cylinder will be the inner cylinder plus the thickness of the concrete on both sides

Radius of total cylinder $= 2.5 + 0.25$

$r_{t o t a l} = 2.75 m$

You can calculate each volume separately and then subtract or create a new formula to do the same thing - both methods will give the same result. As shown in the following calculation:

$V_{c o n c r e t e} = V_{t o t a l} - V_{i n n e r}$

$= [π (r_{t o t a l})^{2} h] - [π (r_{i n n e r})^{2} h]$

$= [π (2.75)^{2} 12] - [π (2.5)^{2} 12]$

$= 285.10 - 235.62$

$= 49.48$

Therefore $49.48 m^{3}$ of concrete would be required to construct the culvert tube.

Check your understanding

Photo of basketball player scoring with a slam dunk.

Portfolio

Now it’s your turn to try a few questions within your notebook. Submit your answers for feedback.

1. Given $x^{6} = 2,500$ , solve for $x$ accurate to two decimal places.

2. The volume of a sphere is given by the formula $V = \frac{4}{3} π r^{3}$ , where r is the radius of the sphere. If the volume of the sphere is 800 cm³, calculate the radius, accurate to two decimal places.

3. The following gumball is to be created with an outer layer of candy and an inner core of bubble gum.

Diagram is a sphere within a sphere. The radius of the inner sphere is labelled rinner and a line segment is drawn from the centre of the sphere to the inner wall of the sphere. The radius of the outer sphere is labelled rtotal and a line segment is drawn from the centre of the sphere to the outer wall of the sphere. — Press here for long description(Open in new window)

If the inner core of this sphere is to have a radius of 0.5 cm and the complete gumball is to have a radius of 1.2 cm, determine the volume of candy required to produce 10,000 of these gumballs. The formula for volume of a sphere is $V = \frac{4}{3} π r^{3}$ .

Conclusion

Congratulations! You have now completed Learning Activity 3.

By working through all of the examples, you have practiced the following concepts:

Solving equations of the form $x^{n} = a$ using rational exponents;
Making connections between formulas and linear, quadratic and exponential equations; and
Solving multi-step problems requiring formulas arising from real-world applications.

Take some time to reflect on this learning activity.

Which level of understanding in the following survey is most reflective of your learning? If you feel that you need more practice, you are encouraged to revisit examples in this learning activity.

Success Criteria	I feel I have mastered the concepts	I feel close to mastering the concepts	I feel I still need to go over additional examples to master the concepts	I’m not quite there yet in mastering the concepts
Solve equations of the form $x^{n} = a$ using rational exponents
Make connections between formulas and linear, quadratic and exponential equations
Solve multi-step problems requiring formulas arising from real-world applications

Assessment Opportunity

Unit 1 Assessment - Mathematical models

This is your first assessment. Complete the Unit 1 Assessment(Opens in a new window).

Feedback and marking

You may receive the following forms of feedback:

Your teacher may highlight the phrases on the rubric that best describe your assignment to show you how you have done.
Your teacher may also provide you with detailed comments about the strengths of your assignment, the areas of the assignment that need improvement, and the steps you should take before submitting another assignment like this one.

Pay careful attention to the following rubric. Your teacher will use it to assess your work. You should refer to it too, so you’ll know exactly what your finished assignment should include.

Success Criteria:

Evaluates exponential expressions (A1)
Simplifies exponential expressions (A1)
Solves exponential equations using technology (A1)
Solves exponential equations using common bases (A1)

Level 4	Level 3	Level 2	Level 1
With a high degree of effectiveness	With considerable effectiveness	With some effectiveness	With limited effectiveness

Success Criteria:

Solves problems involving exponential equations (A1)
Solves multi-step problems (A3)
Calculates rates of change (A2)

Level 4	Level 3	Level 2	Level 1
With a high degree of effectiveness	With considerable effectiveness	With some effectiveness	With limited effectiveness

Success Criteria:

Identifies types of equations (A3)
Identifies rates of change (A2)
Mathematical vocabulary, conventions, symbols and units are used (A3)
Expresses and organizes mathematical thinking clearly and logically (A2)
Reads and interprets an amortization table (A2)

Level 4	Level 3	Level 2	Level 1
With a high degree of effectiveness	With considerable effectiveness	With some effectiveness	With limited effectiveness

Success Criteria:

Solves problems involving exponential equations (A3)
Solves problems involving formulas (A3)

Level 4	Level 3	Level 2	Level 1
With a high degree of effectiveness	With considerable effectiveness	With some effectiveness	With limited effectiveness

The teacher will assess your work using the rubric. Before submitting your assessment, review the rubric to ensure that you are meeting the success criteria to the best of your ability.

Culminating Activity

You will be completing a Culminating Activity for this, which consists of 4 parts. One part will be completed and handed in with each unit.

Part 1 - Researching career opportunities
Part 2 - Personal finance
Part 3 - Geometry and trigonometry
Part 4 - Data management

Assessment Opportunity

Culminating Part 1 - Researching career opportunities

This is an assessment opportunity. Please complete the first part of the Culminating Activity (Opens in new window). Refer to the handout, so you know exactly how your finished assignment should appear.

Feedback and marking

You may receive the following forms of feedback:

Your teacher may highlight the phrases on the rubric that best describe your assignment to show you how you have done.
Your teacher may also provide you with detailed comments about the strengths of your assignment, the areas of the assignment that need improvement, and the steps you should take before submitting another assignment like this one.

Pay careful attention to the following rubric. Your teacher will use it to assess your work. You should refer to it too, so you’ll know exactly what your finished assignment should include.

Success Criteria:

Identifies the occupation, or job title, and the education required to pursue that career (A3)
Identifies the name, and location, of a college program that prepares individuals for success in the chosen career (A3)
Identifies entrance requirements for the chosen program, including high school mathematics courses (A3)
Identifies the salary range or average salary for the chosen career (A3)

Level 4	Level 3	Level 2	Level 1
With a high degree of effectiveness	With considerable effectiveness	With some effectiveness	With limited effectiveness

Success Criteria:

Makes connections between the chosen career and how mathematical modeling relates to the career (A3)
Describes fully an example of mathematical modeling that is explored during the chosen college program (A3)

Level 4	Level 3	Level 2	Level 1
With a high degree of effectiveness	With considerable effectiveness	With some effectiveness	With limited effectiveness

Success Criteria:

Communicates research findings in a clear, coherent manner (A3)
Cites sources fully (A3)
Uses mathematical conventions, vocabulary, and terminology (A3)
Expresses and organizes mathematical thinking (A3)
Expresses and organizes information (A3)
Uses images and links to further reading to enahnce the career inquiry (A3)

Level 4	Level 3	Level 2	Level 1
With a high degree of effectiveness	With considerable effectiveness	With some effectiveness	With limited effectiveness

Success Criteria:

Provides a general job description of the chosen career (A3)
Selects and uses relevant and significant information from research to strengthen the career inquiry (A3)
Makes connections between the chosen college program and the range of career opportunities that might result (A3)
Describes the employment outlook for the chosen career in Canada (A3)

Level 4	Level 3	Level 2	Level 1
With a high degree of effectiveness	With considerable effectiveness	With some effectiveness	With limited effectiveness

The teacher will assess your work using the rubric. Before submitting your assessment, review the rubric to ensure that you are meeting the success criteria to the best of your ability.

Transferable skills survey

Having completed the unit, take the opportunity to review your demonstration of Ontario's Transferable Skills, introduced in 1.1. Complete the Unit 1 Transferable Skills Survey to share your assessment and specific evidence for that rating.

Introduction

Notebook

Join the discussion

What type of equation is it?

Methods for solving

Substituting all given values first

Example 1

Example 2

Step 1

Step 2

Step 3

Step 4

Step 5

Step 6

Step 7

Example 3

Step 1

Step 2

Step 3

Step 4

Step 5

Example

Notebook

Isolate then substitute

Example

Notebook

Solving multi-step problems in real-world applications

Example 1

Example 2

Check your understanding

Portfolio

Conclusion

Assessment Opportunity

Unit 1 Assessment - Mathematical models

Knowledge and Understanding

Thinking

Communication

Application

Culminating Activity

Assessment Opportunity

Culminating Part 1 - Researching career opportunities

Knowledge and Understanding

Thinking

Communication

Application

Transferable skills survey