Functional Modeling Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

easy
A rectangular garden has perimeter 4040 m. Express the area AA as a function of the width ww, find the domain, and determine the width that maximizes area.

Solution

  1. 1
    Perimeter: 2w+2l=40⇒l=20−w2w+2l=40 \Rightarrow l=20-w. Area: A(w)=w⋅l=w(20−w)=20w−w2A(w)=w\cdot l=w(20-w)=20w-w^2.
  2. 2
    Domain: both w>0w>0 and l=20−w>0l=20-w>0, so w∈(0,20)w\in(0,20).
  3. 3
    Maximize: A(w)=−(w2−20w)=−(w−10)2+100A(w)=-(w^2-20w)=-(w-10)^2+100. Maximum area 100100 m² at w=10w=10 m (square garden).

Answer

A(w)=20w−w2A(w)=20w-w^2; domain (0,20)(0,20); maximum 100 m2100\text{ m}^2 at w=10w=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.

About Functional Modeling

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

Learn more about Functional Modeling →

More Functional Modeling Examples

Example 2 medium

A ball is dropped from a [formula] m building. Using [formula], find: (a) height at [formula] s, (b)

Example 3 easy

A taxi charges [formula]2.50[formula][formula] per [formula] mile. Write a cost function [formula] i

Example 4 hard

A cylindrical can must hold [formula] cm³. Express the total surface area [formula] as a function of

← All Functional Modeling Examples Functional Modeling Concept