1. Describe a situation that models a linear pattern and then describe a situation that models a nonlinear pattern. Do not state which situation is linear and which is nonlinear. Your classmates will need to determine which is which and then write a function that models the linear situation. Your classmates will also need to write 1 ordered pair that is a solution to the linear function and explain its meaning in the context of the situation.
A company's cost function is given as C(x) = 20,000 +4.50x, where x is the number of units produced. What is the slope of this function, and what does it represent?
Sure, here are two situations:
Situation A: A person is driving a car on a straight highway at a constant speed. Every hour that passes, the number of miles they have traveled increases by a fixed amount.
Situation B: A person is filling up a water tank using a hose. As time passes, the rate at which the water level rises in the tank changes, sometimes going up faster or slower, depending on various factors.
I won't tell you which situation is linear or nonlinear just yet. I'll leave that to your classmates to figure out!
Situation 1: A situation that models a linear pattern could be tracking the height of a growing plant over time. As the days go by, the plant grows taller by a constant amount each day. The height of the plant can be represented by a linear function.
Situation 2: A situation that models a nonlinear pattern could be calculating the distance traveled by a car over time. Initially, the car starts at rest, but as time increases, the car accelerates, resulting in an increasing rate of distance traveled. In this situation, the distance traveled cannot be represented by a linear function but rather a nonlinear function.
Sure! Here are two situations, one representing a linear pattern and the other representing a nonlinear pattern:
Situation 1: A company sells t-shirts for $10 each. If a customer buys multiple t-shirts, the total cost is directly proportional to the number of t-shirts purchased.
Situation 2: A car is accelerating from rest, and its displacement over time is measured. As time increases, the change in displacement becomes greater and greater.
To determine which situation represents a linear pattern and which represents a nonlinear pattern, we can analyze the relationships between the variables.
In Situation 1, the total cost is proportional to the number of t-shirts purchased. This indicates a linear pattern since the relationship can be represented by a straight line on a graph. The equation for a linear relationship is typically in the form of y = mx + b, where y represents the total cost, x represents the number of t-shirts, m represents the constant rate (price per t-shirt), and b represents any additional fixed costs.
In Situation 2, the change in displacement of the car is increasing with time. This indicates a nonlinear pattern as the relationship does not follow a straight line. Nonlinear relationships can be modeled using various equations, such as quadratic, exponential, or logarithmic functions, depending on the specific behavior observed.
Now, let's focus on Situation 1 and model it using a linear function.
To write the linear function for Situation 1:
Let's assign y as the total cost and x as the number of t-shirts purchased.
y = mx + b
In this case, the constant rate, m, represents the cost per t-shirt, which is $10. Therefore, the equation becomes:
y = 10x + b
The additional fixed cost, b, will vary depending on the situation. For example, if the b value is $5, the equation becomes:
y = 10x + 5
Now, let's find one ordered pair that is a solution to this linear function and explain its meaning in the context of Situation 1.
Let's consider if a customer buys 3 t-shirts. Plugging this value into the equation, we can solve for y:
y = 10 * 3 + 5
y = 30 + 5
y = 35
Therefore, the ordered pair (3, 35) represents a solution to the linear function in this context. It means that if a customer purchases 3 t-shirts, the total cost would be $35.
I hope this helps! If you have any further questions or need clarification, feel free to ask.