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Identify and explain whether the following quantities change at a constant

rate relative to each other.
• the number of bacteria in a culture that doubles every hour
• the population of a city that is growing at a rate of 2% each year
• the distance traveled by a car going 60 miles per hour
• the amount of money earned at $15.75 per hour
B. Write a mathematical equation to model the relationship between two
quantities that change at a constant rate relative to each other.
C. Use your equations from part B to solve a real-world problem.
D. Explain how your equation can be used to make predictions about the future
values of the two quantities.
E. Identify and explain any limitations of your equation.

9 answers

A. Identifying Changes in Rates

  1. Number of bacteria in a culture that doubles every hour:

    • This quantity does not change at a constant rate; it changes exponentially. Specifically, if we start with \( N_0 \) bacteria, after \( t \) hours, the number will be \( N = N_0 \times 2^t \).

  2. Population of a city growing at a rate of 2% each year:

    • This quantity also does not change at a constant rate; it increases exponentially. If the current population is \( P_0 \), then after \( t \) years, the population will be \( P = P_0 \times (1 + 0.02)^t \).

  3. Distance traveled by a car going 60 miles per hour:

    • This quantity changes at a constant rate. The relationship between distance \( D \), speed \( S \), and time \( t \) is linear, given by \( D = S \times t \). Here, \( D = 60t \).

  4. Amount of money earned at $15.75 per hour:

    • This quantity also changes at a constant rate. If \( E \) is the total earnings and \( h \) is the number of hours worked, the relationship is \( E = 15.75h \).

In summary, the last two quantities (distance traveled and money earned) change at a constant rate relative to each other, while the bacteria culture and city population grow at exponentially changing rates.

B. Mathematical Equations for Constant Rates

  1. Distance traveled by a car: \( D = 60t \)
  2. Amount of money earned: \( E = 15.75h \)

Setting the two quantities equal (assuming 1 hour of travel corresponds with 1 hour of work for simplicity):

Let \( t = h \):

  • Distance traveled = Amount earned: \[ 60t = 15.75t \]

C. Real-World Problem Using Equations

Suppose a car travels at 60 miles per hour for 3 hours and a worker earns $15.75 per hour for the same duration. How far does the car travel, and how much does the worker earn?

Using the equations:

  1. For distance: \[ D = 60 \times 3 = 180 \text{ miles} \]
  2. For earnings: \[ E = 15.75 \times 3 = 47.25 \text{ dollars} \]

Thus, after 3 hours, the car travels 180 miles, and the worker earns $47.25.

D. Making Predictions About Future Values

The equations \( D = 60t \) and \( E = 15.75h \) can predict values for any time \( t = h \). For instance, if we want to know how far the car travels and how much the worker earns after 5 hours:

  1. Distance: \[ D = 60 \times 5 = 300 \text{ miles} \]
  2. Earnings: \[ E = 15.75 \times 5 = 78.75 \text{ dollars} \]

E. Limitations of the Equation

  1. Assumption of Constant Rate: The equations assume speed and hourly wage remain constant. In reality, speed may vary due to traffic, and hourly wages might change (due to raises or overtime).

  2. No External Factors Considered: These equations do not consider external factors like stops, breaks, inflation affecting wages, or variations in route taken by the car.

  3. Linear Context: They only apply in contexts where the relationships being modeled are linear. They would not hold true for scenarios where acceleration or variable pay rates come into play.

  4. Time Frame Alignment: The equations assume that the time spent driving equals the time spent working, limiting their application in situations where these activities do not occur simultaneously.

In conclusion, while these models provide a useful approximation under certain conditions, their applicability may be limited by variations in real life.

can you make it more simple and like a kid wrote it

Sure! Here’s a simpler version as if a kid wrote it:

A. Do These Things Change at the Same Rate?

  1. Bacteria in a culture:

    • Nope! The bacteria double every hour, so one hour there are 10, the next hour there are 20, then 40, then 80, and it keeps getting bigger and bigger super fast.

  2. City population (growing by 2%):

    • Nope! The population grows but not at a steady pace. If there are 100 people this year, next year it’ll be 102, then 104.04, and so on. It keeps getting bigger but not by the same amount every year.

  3. Distance traveled by a car:

    • Yes! If a car goes 60 miles per hour, then if it drives for 1 hour, it goes 60 miles; for 2 hours, it goes 120 miles. It’s a straight line that keeps adding the same amount of miles!

  4. Money earned at $15.75 per hour:

    • Yes! If someone works for 1 hour, they earn $15.75; for 2 hours, they earn $31.50. They're earning the same amount every hour, just like the miles!

B. Simple Math Equations

  1. Distance traveled by a car: \[ D = 60t \] (where \( D \) is distance and \( t \) is time in hours)

  2. Money earned: \[ E = 15.75h \] (where \( E \) is money and \( h \) is hours worked)

C. Real-World Example

Let’s say the car drives for 3 hours and someone works for 3 hours.

  1. Distance: \[ D = 60 \times 3 = 180 \text{ miles} \]

  2. Money earned: \[ E = 15.75 \times 3 = 47.25 \text{ dollars} \]

So the car drives 180 miles, and the person earns $47.25.

D. Making Predictions

We can use our equations to guess what will happen in the future!

For example, if the car drives for 5 hours and the worker works for 5 hours:

  1. Distance: \[ D = 60 \times 5 = 300 \text{ miles} \]

  2. Money earned: \[ E = 15.75 \times 5 = 78.75 \text{ dollars} \]

The car would go 300 miles, and the worker would earn $78.75.

E. What’s Not So Great About These Equations

  1. Always the Same Speed/Wage:

    • The equations assume the car always goes 60 miles per hour and the worker always earns $15.75. What if there’s traffic or the worker gets a raise?

  2. No Breaks:

    • They don’t think about breaks or stops. What if the car has to stop for gas?

  3. Only Works for This Situation:

    • These math rules only work for this kind of situation. If the car speeds up or slows down, it won’t work.

So, our equations are super helpful but have some limits, just like we all do!

can you make it into a small paragraph

Sure! Here’s a small paragraph summarizing everything:

The number of bacteria in a culture and the city population grow at different rates; the bacteria double every hour, while the population increases by 2% each year, so they don't change steadily. However, the distance a car travels at 60 miles per hour and the money someone earns at $15.75 per hour both change at a constant rate. For example, after 3 hours, the car goes 180 miles, and the person earns $47.25. We can use the equations \( D = 60t \) for distance and \( E = 15.75h \) for money to predict future values. For instance, if they both work or travel for 5 hours, the car will go 300 miles, and the worker will earn $78.75. These equations are helpful, but they assume the speed and pay stay the same, and they don’t account for breaks or changes!

can you make it shorter

Sure! Here’s a shorter paragraph:

Bacteria and city populations grow at different rates—bacteria double every hour, while city populations increase by 2% each year. In contrast, the distance a car travels at 60 miles per hour and the money earned at $15.75 per hour both change steadily. For example, after 3 hours, the car goes 180 miles, and the person earns $47.25. We can use the equations \( D = 60t \) and \( E = 15.75h \) to predict future values, like traveling 300 miles or earning $78.75 after 5 hours. However, these equations assume constant speed and pay, not accounting for breaks or changes.

sorry I keep asking this but can you make shorter

Of course! Here’s an even shorter version:

Bacteria double every hour, and city populations grow by 2% each year, so they don't change steadily. However, the distance a car travels at 60 miles per hour and the money earned at $15.75 per hour both increase steadily. For example, after 3 hours, the car goes 180 miles, and the worker earns $47.25. We can use the equations \( D = 60t \) and \( E = 15.75h \) to predict future amounts, but these assume constant speed and pay without breaks.