1 Mathematical Modeling and Quantitative Reasoning Free Online FlashCards

What is a mathematical model?

A mathematical model is a simplified representation of a real-world situation using quantities, symbols, equations, functions, graphs, tables, or diagrams.

How do input and output variables differ?

An input, or independent variable, is used to explain or determine an output, or dependent variable. If distance depends on time, it can be written as $d(t)$.

What model describes the draining tank?

The tank model is $V(t)=500-4t$, where $t$ is time in minutes and $V$ is volume in gallons.

What is an assumption in modeling?

An assumption is a statement accepted as approximately true to make a real-world model workable. Inaccurate assumptions can make predictions biased or unreliable.

What is a constraint?

A constraint restricts the values allowed by the situation. For the tank, the meaningful domain is $0\le t\le125$ minutes because time and water volume cannot be negative.

How do units check an equation?

Units are dimensionally consistent when the quantities being combined have compatible units. For example, $(4\text{ gallons per minute})(t\text{ minutes})$ gives $4t$ gallons.

What is the cyclist's average speed?

The cyclist's average speed is $\frac{18\text{ miles}}{1.5\text{ hours}}=12\text{ miles per hour}$. A unit must accompany the numerical answer.

What distance does 3 inches represent on this map?

A map scale of $1\text{ inch}=20\text{ miles}$ means that 3 inches represent $3(20)=60$ miles.

What process translates a situation into mathematics?

Define variables with meanings and units, identify the relationship, state assumptions and constraints, choose a representation, check units and reasonableness, and interpret the result in context.

What is the subscription cost model?

The subscription model is $C(n)=12+3n$, where $n$ is the number of premium movies and $C$ is the monthly cost in dollars.

What values can the movie variable $n$ take?

Because $n$ counts movies, its allowed values are the nonnegative whole numbers: $n\in\{0,1,2,3,\ldots\}$.

What is $C(5)$ for the subscription model?

Using $C(n)=12+3n$, $C(5)=12+3(5)=27$. The monthly cost is $27 under the stated assumptions.