Mathematical Modeling

Identify variables and fixed parameters, then translate real-world situations into mathematical models.

Student Task

For each scenario:

  1. Identify the independent variable.
  2. Identify the dependent variable(s).
  3. Identify the fixed parameters.
  4. Build the mathematical model.
Translation tip: “per unit” usually becomes multiplication by x; “flat fee” or “starting amount” is usually a fixed parameter; “total” combines the relevant terms.

Original Delivery Van Example

A local delivery business rents a van for ₱8,000 per day. Fuel and maintenance cost ₱9 per kilometer. The business charges ₱80 per kilometer.

ElementAnswer
Independent variablex = kilometers driven
Dependent variableP(x) = daily profit
Fixed parameter₱8,000 daily rental
Variable parameters₱9/km operating cost; ₱80/km delivery fee
ModelP(x) = 80x − (8,000 + 9x) = 71x − 8,000

At 250 km: P(250) = ₱9,750.

Break-even: 71x − 8,000 = 0, so x = 112.68 km, or about 113 km.

Important Modeling Note: One or More Independent Variables

For the introductory examples, the app uses one independent variable, x. This makes the idea of a function easier to learn.

Some real-world models, however, have more than one independent variable. For example, a delivery company's daily revenue might depend on both the number of deliveries and the average distance per delivery:

R(x,y) = 80xy

Here, x = number of deliveries and y = average kilometers per delivery. Revenue is a function of two independent variables.

Similarly, total delivery cost could be:

C(x,y) = 8,000 + 9xy

and profit could be:

P(x,y) = R(x,y) − C(x,y) = 80xy − (8,000 + 9xy)

Explore and Modify the Models

Select an example, change its values, and click Update Model. The mathematical model and numerical result will change.

Extension: Model With Two Independent Variables

A delivery service charges ₱80 per kilometer. Suppose the number of deliveries is x and the average distance per delivery is y.

Practice: Identify Before Modeling

Select the best answer. Feedback appears immediately.

Complete Solution Key

Use after students attempt the activities.

All Philippine monetary amounts are illustrative teaching examples, not official prices or statistics.