Objective 1 - Identify Type of Model

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Note: There are no textbook or videos directly to this section. If you want to review a certain type of model, you will need to go back to that Module.

We summarize the types of models we’ve looked at below.

Linear Model:

Used when we have a constant variation between two quantities.
$y = mx + b$ . Can be multiple lines added together.
Phrases to look for: consant, steadily increasing/decreasing, adding/subtracting [/, /,] every [/, /,].

Direct Variation:

Used when we have a direct variation between two quantities (as one quantity increases, the other increases).
$y = kx^n$ . Joint variation may have more than one variable (like $y = kx^n z^m$ )
Phrases to look for: vary directly, directly proportional, “as one increases, so does the other”.

Inverse Variation:

Used when we have an indirect variation between two quantities (as one quantity increases, the other decreases).
$y = \frac {k}{x^n}$ . Combined with joint variation, there may be more than one variable (like $PV=nRT$ ).
Phrases to look for: vary indirectly, directly proportional, “as one increases, the other decreases”.

Logarithmic Model:

Used when we have a rapid early growth, then slower growth later.
$y = \log (kx)$ . Remember that $\ln (x) = \log _e(x)$ .
Phrases to look for: rapid early growth/decay, no bound on growth/decay.

Exponential Model:

Used when we have a slow initial growth, then rapid growth later.
$y = a^{kx}$ . Common bases are 2, 3, and $e$ .
Phrases to look for: rapid late growth/decay, bounds on growth/decay.

Determine the type of model that would be most appropriate for each situation below. Answers will be either:

Linear
Direct
Indirect
Logarithmic
Exponential
General (if we are going to use the general form of a particular function)

Your bank offers a savings account that will increase your total balance by 0.2% annually. You want to decide how much to initially deposit and if the initial deposit makes a big difference in the long run. S

$\answer [format=string]{Exponential}$

A ball is dropped from the top of Century Tower. The ball steadily picks up speed before hitting the ground. You want to figure out what the ball’s height is at a certain time.

$\answer [format=string]{Direct}$

Chemists commonly create a solution by mixing two products of differing concentrations together. For example, a chemist could have large amounts of a 10% acid solution and a 30% acid solution, but need a 10 liter 15% solution.

$\answer [format=string]{Linear}$

[Astronomy] Kepler’s Third Law: The square of the time, $T$ , required for a planet to orbit the Sun is directly proportional to the cube of the mean distance, $a$ , that the planet is from the Sun.

$\answer [format=string]{Direct}$

[Physics] The rate of vibration of a string under constant tension, $r$ , varies inversely with the length of the string, $l$ .

$\answer [format=string]{Indirect}$

A population of bacteria doubles every hour.

$\answer [format=string]{Exponential}$

[Anthropology] Radiocarbon dating is used to calculate the approximate date a plant or animal died by noting the percentage of carbon-14, $r$ in the object. The age of the object $t$ , in years, is directly proportional to the natural log of the percentage of carbon-14, $r$ in the object.

$\answer [format=string]{Logarithmic}$

The weight of an object above the surface of Earth varies inversely with the square of the distance from the center of Earth.

$\answer [format=string]{Indirect}$

The kinetic energy $K$ of a moving object varies jointly with its mass $m$ and the square of its velocity $v$ .

$\answer [format=string]{Direct}$

Carlos has taken an initial dose of a prescription medication orally. The medicine is absorbed rapidly by the large intestine and absorbed slowly as it is digested otherwise.

$\answer [format=string]{Exponential}$

$\answer [format=string]{Logarithmic}$

A ball is dropped from the top of Century Tower. The ball steadily picks up speed before hitting the ground. You want to figure out what the ball’s speed is at a certain time.

$\answer [format=string]{Linear}$

Figure 1: A sigmodial curve.

$\answer [format=string]{Exponential}$

Kappa Delta is hosting an all-you-can-eat pancake fundraiser to support the prevention of child abuse. Adult (18+) tickets are $10 and teen (10-17) tickets are $5. Children under 10 are let in without a ticket. The ticket-sellers only keep track of the total number of tickets sold and total revenue, but want to know how many adult and teen tickets were sold.

$\answer [format=string]{Linear}$