Activity
For numbers 1 - 4, use the situation below:
After a parade, a group of volunteers is helping to pick up the trash along a 2-mile stretch of a road. The group decides to divide the length of the road so that each volunteer is responsible for cleaning up equal-length sections.
Find the length of a road section for each volunteer if there are the following numbers of volunteers. Be prepared to show your reasoning.
8 volunteers
Compare your answer:
mile
10 volunteers
Compare your answer:
mile
25 volunteers
Compare your answer:
mile or 0.08 mile
36 volunteers
Compare your answer:
or mile
Write an equation that would make it easy to find , the length of a road section in miles for each volunteer, if there are volunteers.
Compare your answer:
For numbers 6 - 9, find the number of volunteers in the group if each volunteer cleans up a section of the following lengths. Be prepared to show your reasoning.
0.4 mile
Compare your answer:
5 volunteers
mile
Compare your answer:
7 volunteers
0.125 mile
Compare your answer:
16 volunteers
mile
Compare your answer:
15 volunteers
Write an equation that would make it easy to find the number of volunteers, , if each volunteer cleans up a section that is miles.
Compare your answer:
Are you ready for more?
Extending Your Thinking
Let's think about the graph of the equation .
Create a list pairs that will help you graph the equation. Make sure to include some negative numbers for and some numbers that are not integers.
Compare your answer:
Your answer may vary, but here is an example:
Use the graphing tool or technology outside the course. Graph the equation that represents this scenario using the Desmos tool above.
Compare your answer:
What do you think the graph looks like when is between 0 and ? Continue to use the graphing tool or technology outside the course to graph the function with your points. Try some values of to test your idea. (Students were provided access to Desmos.)
Compare your answer:
Your answer may vary, but here is an example:
The graph gets steeper and steeper as gets closer to 0.
What is the largest value can ever be?
with your points. Try some values of to test your idea. In Desmos, zoom in on your graph.
Compare your answer:
Your answer may vary, but here is an example:
There is no limit to the value of 𝑦.
Self Check
Additional Resources
Solve a Formula for a Specific Variable
We have all probably worked with some geometric formulas in our study of mathematics. Formulas are used in many fields, so it is important to recognize formulas and be able to manipulate them easily.
It is often helpful to solve a formula for a specific variable. If you need to put a formula in a spreadsheet, it is not unusual to have to solve it for a specific variable first. We isolate that variable on one side of the equal sign with a coefficient of 1, and all other variables and constants are on the other side of the equal sign.
Geometric formulas often need to be solved for another variable, too. The formula is used to find the volume of a right circular cone when given the radius of the base and height. In the next example, we will solve this formula for the height.
Example 1
Solve for .
Step 1 - Remove the fraction on the right.
Multiply both sides by 3.
Step 2 - Simplify.
Step 3 - Divide both sides by .
We could now use this formula to find the height of a right circular cone when we know the volume and the radius of the base, by using the formula
In the sciences, we often need to change temperature from Fahrenheit to Celsius or vice versa. If you travel in a foreign country, you may want to change the Celsius temperature to the more familiar Fahrenheit temperature.
Example 2
Solve
for .
Step 1 - Remove the fraction on the right.
Multiply both sides by 9/5.
Step 2 - Simplify.
Step 3 - Add 32 to both sides.
Continue explanation as is
Example 3
Solve the formula for .
Step 1 - Subtract 8x from both sides to isolate the term with .
Step 2 - Simplify.
Step 3 - Divide both sides by 7 to make the coefficient of equal to 1.
Step 4 - Simplify.
Now, try to solve for .
Answer:
Step 1 - Remove the fraction on the right.
Multiply by 2.
Step 2 - Simplify.
Step 3 - Divide both sides by .
Step 4 - Simplify.
Step 5 - Subtract from both sides to solve for v0.
Answer:
Try it
Try It: Solve a Formula for a Specific Variable
Solve for .
Here is how to solve this equation for .
Step 1 - Subtract from each side.
Step 2 - Simplify.
Step 3 - Divide both sides by 8.
The equation solved for is .