Math 8 4 9 Homework Help Morgan
TLDRThis Math 8 Unit 4 lesson explores equations to compare different scenarios, such as babysitting rates, water tank levels, elevator heights, and cell phone plans. Students learn to set up equations, solve for variables, and determine when two quantities are equal, highlighting the importance of understanding rates and changes over time.
Takeaways
- 🧮 The lesson focuses on comparing different situations using equations to determine when they yield the same result.
- 👶 In the babysitting scenario, two payment options are compared: one with a flat rate for the first hour and a variable rate for additional hours, and another with a higher initial rate and a lower rate for additional hours.
- 💡 The key to solving these scenarios is to set up equations and solve for the variable that makes the two situations equal.
- 🕒 The babysitting example shows that after five hours, both payment options result in the same earnings, with the first option being more beneficial before five hours and the second option after.
- 🚰 The water tank activity involves comparing the rates at which water levels change in two tanks, one increasing and one decreasing, to estimate when they will have the same water level.
- 📊 The water tank example uses a table to estimate the time when both tanks have a similar water level, highlighting the importance of visual data analysis.
- 📈 The rate of water increase in tank one is 30 liters per minute, and the rate of decrease in tank two is 20 liters per minute, which helps in setting up the equation to find when they equalize.
- 🏢 The elevator activity compares the travel times of two elevators moving in opposite directions, with one starting above ground and the other below.
- ⏱ The elevator problem involves solving for the time it takes for each elevator to reach ground level and when they would pass each other.
- 📞 The cell phone plan comparison determines the breakeven point in months where the total cost of two different plans becomes equal, considering the cost of the phone with one of the plans.
- 🚴♂️ The biking scenario involves two individuals biking in the same direction at different speeds, with one starting earlier, and solving for the time when they meet.
- 🔢 The lesson includes various mathematical problems that require setting up and solving equations to find when certain expressions or situations are equivalent.
Q & A
What is the main topic of Math 8 Unit 4 Lesson 9?
-The main topic of Math 8 Unit 4 Lesson 9 is 'When are they the same,' which focuses on using equations to compare different situations and determine when they yield the same result.
In the babysitting scenario, what are the two different payment options presented?
-The two payment options are: Option 1 charges $5 for the first hour and $8 for each additional hour. Option 2 charges $15 for the first hour and $6 for each additional hour.
How many hours of babysitting would make the two payment options equal in terms of earnings?
-After 5 hours of babysitting, both payment options would yield the same amount of earnings.
What is the purpose of setting the two babysitting equations equal to each other?
-Setting the two equations equal helps to find out at what point in time (number of hours worked) the earnings from both options would be the same.
In the water tanks activity, how are the amounts of water changing in each tank?
-In Tank 1, the amount of water is increasing, while in Tank 2, it is decreasing.
What is the estimated time when the water levels in both tanks would be the same?
-The estimated time when the water levels in both tanks would be the same is between 15 and 20 minutes, likely closer to 20 minutes.
What is the rate of increase in Tank 1 in liters per minute?
-Tank 1 is increasing at a rate of approximately 30 liters per minute.
What is the rate of decrease in Tank 2 in liters per minute?
-Tank 2 is decreasing at a rate of 20 liters per minute.
How can you determine when the two water tanks have the same amount of water mathematically?
-You can determine when the two water tanks have the same amount of water by setting the equations representing the water levels in each tank equal to each other and solving for time (t).
What is the mathematical formula for the water level in Tank 1 after t minutes?
-The formula for the water level in Tank 1 after t minutes is 30t + 25 liters.
What is the mathematical formula for the water level in Tank 2 after t minutes?
-The formula for the water level in Tank 2 after t minutes is -20t + 1000 liters.
At what time (in minutes) will the water levels in both tanks be equal?
-The water levels in both tanks will be equal at 19.5 minutes.
In the elevator scenario, what is the height of each elevator when they are at the same level?
-The height at which the elevators are at the same level is -2.5 meters, which means 2.5 meters below ground level.
How long would it take for each elevator to reach ground level from their respective starting points?
-It would take Elevator A 16 seconds and Elevator B 12 seconds to reach ground level from their starting points.
If two elevators are traveling towards each other, how can you calculate the time it takes for them to pass each other?
-To calculate the time it takes for the elevators to pass each other, set the equations representing their heights over time equal to each other and solve for time (h).
Which elevator would reach an underground parking lot 14 meters below ground first, and how long would it take?
-Elevator A would reach the underground parking lot first, taking 4.8 seconds, while Elevator B would take 23.2 seconds.
In the cell phone plan comparison, how many months would it take for the cost of Plan B (without a free phone) to equal the cost of Plan A (with a free phone)?
-It would take 25 months for the cost of Plan B to equal the cost of Plan A when considering the initial cost of the phone.
In the biking scenario, how long would it take for Han and Priya to meet if Han has been biking for 't' hours?
-Han and Priya would meet after 1.5 hours of Han biking, considering Priya started half an hour earlier.
What value of 'x' makes the expressions (2/3)x + 2 and (4/3)x - 6 have the same value?
-The value of 'x' that makes the expressions equal is 12.
How can you determine if an equation is true for all values, one value, or no values of 'x'?
-You can determine this by analyzing the structure of the equation and solving it to see if it results in a specific value, any value, or no value for 'x'.
In the rigid transformation problem, what type of transformation would align shape A with shape B without changing their dimensions?
-A 180-degree rotation around the origin (center point at zero, zero) would align shape A with shape B without changing their dimensions.
Outlines
💼 Babysitting Pay Comparison
In this segment, the video discusses a scenario where one must choose between two babysitting payment options. The first option is charging $5 for the first hour and $8 for each additional hour, while the second option is charging $15 for the first hour and $6 for each additional hour. The lesson focuses on determining when these two payment methods yield the same earnings. By setting up equations for each scenario and solving for the number of hours (h), it's found that after 5 hours, the earnings from both methods would be equal. Before 5 hours, the $15 initial charge would yield more earnings, but after 5 hours, the $8 hourly rate would result in higher earnings.
🌊 Water Tank Levels Analysis
This paragraph explores the concept of water tank levels using two different tanks. Tank one starts with 25 liters and increases, while tank two starts with 1000 liters and decreases. The video uses a table to estimate when the tanks will have the same amount of water, which is predicted to be between 15 and 20 minutes, likely closer to 20 minutes. The increase in tank one is calculated to be 30 liters per minute, and the decrease in tank two is 20 liters per minute. The equations for the water levels in each tank are derived, and it's determined that at 19.5 minutes, the water levels in both tanks will be equal.
🏢 Elevator Height and Meeting Points
The video script introduces a scenario with two elevators in a building, one moving above ground and the other below. The time it takes for each elevator to reach a certain height is given, and equations are set up to find the height of each elevator at a specific time. Elevator A is found to be at -20 meters (below ground) and Elevator B at 15 meters (above ground). The time it takes for each elevator to reach ground level is calculated, with Elevator A taking 16 seconds and Elevator B taking 12 seconds. The script also explores when and where the elevators would pass each other, concluding that they would meet at -2.5 meters. Lastly, it discusses which elevator would reach an underground parking lot first, determining that Elevator A would do so in the shortest time.
📱 Cell Phone Plan Cost Comparison
This segment compares two cell phone plans. Plan A costs $70 per month and includes a free phone, while Plan B costs $50 per month without a phone, but requires an initial $500 phone purchase. The video calculates the breakeven point, or the number of months it takes for the total cost of both plans to be equal. The calculation shows that it would take 25 months for the costs to be the same. Additional examples include calculating the distance traveled by two bikers moving at different speeds and determining when they would meet, as well as solving equations to find values that make certain expressions equal.
🔢 Equations and Transformations
The final paragraph covers solving various equations and understanding rigid transformations. It starts with solving equations for specific variables, such as finding the value of 'd' in an equation involving 'd' and 'a', and then solving for 'k' in another equation. The paragraph also involves solving an equation involving 'y' and transforming it to find the solution. The last part discusses rigid transformations, specifically rotations, and how a 180-degree rotation around the origin can align shapes in a specific way.
Mindmap
Keywords
💡Equation
💡Babysitting Charges
💡Water Tanks
💡Rigid Transformation
💡Elevator Travel Time
💡Cell Phone Plan
💡Biking Speed
💡Breakeven Point
💡Solving Equations
💡Rote Learning
💡Graphical Representation
Highlights
Introduction to the concept of 'when are they the same' using equations to compare different situations.
Babysitting scenario: Comparing two payment methods to determine when they yield the same earnings.
Equation setup for babysitting scenario: 5 + 8h vs. 15 + 6h, solving for h to find when earnings are equal.
Result from babysitting scenario: After 5 hours, both payment methods yield the same earnings.
Water tanks activity: Comparing water levels in two tanks to estimate when they will be equal.
Tank one increases by 30 liters per minute, tank two decreases by 20 liters per minute.
Estimation of equal water levels in tanks between 15 and 20 minutes.
Equation for water in tank one: 30t + 25, and for tank two: -20t + 1000.
Solving the equations for water tanks to find the time when they have equal water levels.
Result from water tanks: Equal water levels at 19.5 minutes.
Elevator scenario: Comparing travel times and heights for two elevators.
Equations for elevator heights: 0.8h + 16 for elevator A, -0.8h + 12 for elevator B.
Calculating the time it takes for each elevator to reach ground level.
Determining which elevator reaches an underground parking lot first.
Cell phone plan comparison: Deciding when the cost of two different plans becomes equal.
Equation setup for cell phone plans: 70 - 50m = 500, solving for m.
Result from cell phone plans: It takes 25 months for the costs to be the same.
Biking scenario: Writing expressions for distance traveled by two bikers and finding when they meet.
Equation for meeting point in biking scenario: 16t = 12(t + 0.5), solving for t.
Result from biking scenario: They meet after 2 hours.
Solving equations to match stories and find values of x.
Determining when two expressions have the same value using algebraic manipulation.
Solving equations for all values of x, one value, or no values.
Using rigid transformation to align shapes by rotating 180 degrees around the origin.
Transcripts
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