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Problem 104

# $$\nabla$$ Describe a real-life situation in which a quadratic model would be more appropriate than an exponential model and one in which an exponential model would be more appropriate than a quadratic model.

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A quadratic model is more appropriate for modeling the trajectory of a projectile, as its parabolic shape represents the object's motion affected by gravity, initial velocity, and constant acceleration. The equation is $$y(x) = ax^2 + bx + c$$. An exponential model is more suitable for population growth, such as bacteria, where the population increases rapidly with each generation. The equation is $$y(x) = A \cdot b^x$$, where A is the initial population and b is the growth factor.
See the step by step solution

## Step 1: Quadratic Model Situation

A real-life situation where a quadratic model would be more appropriate than an exponential model is the trajectory of a projectile. When an object is thrown or launched, its motion can be modeled using a quadratic equation, with the height (y) being a function of time (x). The equation usually takes the form: $$y(x) = ax^2 + bx + c$$ where a, b, and c are constants. The parabolic shape of a quadratic model is ideal for representing the projectile's motion as it represents the object's initial position, the combined effect of gravity and the object's initial velocity, and the constant acceleration due to gravity.

## Step 2: Exponential Model Situation

A real-life situation where an exponential model would be more appropriate than a quadratic model is population growth. Populations of certain species (for example bacteria) often grow exponentially, with the number of individuals (y) as a function of time (x). The equation usually takes the form: $$y(x) = A \cdot b^x$$ where A is the initial population and b represents the growth factor. Exponential growth models can capture the continuous increase in population with each generation due to rapid reproduction. This type of behaviour cannot be accurately represented using a quadratic model as the population would eventually reach its maximum and start decreasing, which is not the case in many population growth scenarios.

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