Functional Modeling Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Functional Modeling.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Functional modeling uses functions to represent relationships between real-world quantities — choosing the right function family to capture the observed pattern.

Translate a situation into a function, then use math to analyze it.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Models simplify reality while capturing essential relationships.

Common stuck point: Fitting any function family to data is possible — the question is whether the chosen family matches the underlying mechanism, not just the observed data points.

Sense of Study hint: Start by identifying the variables and asking: is the relationship linear, quadratic, or exponential? Plot the data and look at the shape.

Worked Examples

Example 1

easy
A rectangular garden has perimeter 40 m. Express the area A as a function of the width w, find the domain, and determine the width that maximizes area.

Solution

  1. 1
    Perimeter: 2w+2l=40 \Rightarrow l=20-w. Area: A(w)=w\cdot l=w(20-w)=20w-w^2.
  2. 2
    Domain: both w>0 and l=20-w>0, so w\in(0,20).
  3. 3
    Maximize: A(w)=-(w^2-20w)=-(w-10)^2+100. Maximum area 100 m² at w=10 m (square garden).

Answer

A(w)=20w-w^2; domain (0,20); maximum 100\text{ m}^2 at w=10 m
Functional modeling turns a geometric constraint (fixed perimeter) into an algebraic function. The optimal shape — a square — emerges naturally from completing the square on the area function.

Example 2

medium
A ball is dropped from a 100 m building. Using h(t)=100-4.9t^2, find: (a) height at t=3 s, (b) time to hit the ground, (c) interpret h'(t) at impact.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
A taxi charges \2.50 base fare plus \0.40 per \frac{1}{4} mile. Write a cost function C(m) in terms of miles m and find the cost of a 6-mile ride.

Example 2

hard
A cylindrical can must hold 500 cm³. Express the total surface area S as a function of radius r, and find the value of r that minimizes material use.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

function definitionmodeling with equations