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 β€” 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 β†’

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: Functional modeling turns a real situation into a function by choosing the function type whose behavior matches the data.

Common stuck point: The procedure for functional modeling is the easy part; the trap is forcing a line onto exponential data. Asking "Are you choosing and building a function to represent a real-world relationship?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Are you choosing and building a function to represent a real-world relationship?

Worked Examples

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.

Answer

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

First step

1
Perimeter: 2w+2l=40β‡’l=20βˆ’w2w+2l=40 \Rightarrow l=20-w. Area: A(w)=wβ‹…l=w(20βˆ’w)=20wβˆ’w2A(w)=w\cdot l=w(20-w)=20w-w^2.

Full solution

  1. 2
    Domain: both w>0w>0 and l=20βˆ’w>0l=20-w>0, so w∈(0,20)w\in(0,20).
  2. 3
    Maximize: A(w)=βˆ’(w2βˆ’20w)=βˆ’(wβˆ’10)2+100A(w)=-(w^2-20w)=-(w-10)^2+100. Maximum area 100100 mΒ² at w=10w=10 m (square garden).
Functional modeling turns a geometric constraint (fixed perimeter) into an algebraic function. The optimal shape β€” a square β€” emerges naturally from completing the square on the area function.

Example 2

medium
A ball is dropped from a 100100 m building. Using h(t)=100βˆ’4.9t2h(t)=100-4.9t^2, find: (a) height at t=3t=3 s, (b) time to hit the ground, (c) interpret hβ€²(t)h'(t) at impact.

Example 3

medium
A printer charges a setup fee plus per-page fee. 50 pages costs \$22; 100 pages costs \$32. Find the linear cost function.

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\$2.50 base fare plus $0.40\$0.40 per 14\frac{1}{4} mile. Write a cost function C(m)C(m) in terms of miles mm and find the cost of a 66-mile ride.

Example 2

hard
A cylindrical can must hold 500500 cmΒ³. Express the total surface area SS as a function of radius rr, and find the value of rr that minimizes material use.

Example 3

easy
A taxi charges a $3 flat fee plus $2 per mile. Write the cost CC as a function of miles mm.

Example 4

easy
A bacteria count doubles every hour, starting at 100. Write the count NN as a function of hours tt.

Example 5

easy
A rectangle has a fixed perimeter of 20. Write its area AA as a function of width ww.

Example 6

easy
A phone plan costs $40 per month with no per-minute charge. Write monthly cost CC as a function of minutes mm.

Example 7

easy
Water drains from a 50-liter tank at 5 liters per minute. Write the volume VV as a function of minutes tt.

Example 8

easy
A function f(x)=2xf(x) = 2x models dollars earned per hour worked, xx. What is a sensible domain restriction for this real-world model?

Example 9

easy
A square's area as a function of side length ss is what?

Example 10

easy
A trend in data shows points curving sharply upward, not on a straight line. Which model family is more appropriate: linear or exponential?

Example 11

medium
A ball is thrown up; its height is h(t)=βˆ’5t2+20th(t) = -5t^2 + 20t meters. Find when it lands (height returns to 0) and state why the model is invalid afterward.

Example 12

medium
A population is 200 at year 0 and 800 at year 2, growing exponentially. Find the model P(t)=P0btP(t) = P_0 b^t.

Example 13

medium
A company's profit is revenue minus cost: revenue R(x)=10xR(x) = 10x and cost C(x)=4x+120C(x) = 4x + 120 for xx units. Write profit P(x)P(x) and find the break-even point.

Example 14

medium
A car depreciates 20% per year from $30000. Write its value V(t)V(t) and find its value after 2 years.

Example 15

medium
A farmer has 100 m of fencing for a rectangular pen against a straight wall (only 3 sides fenced). Write the enclosed area as a function of the side xx perpendicular to the wall.

Example 16

medium
A model predicts f(x)=3x+1f(x) = 3x + 1 thousand sales for ad spend xx (thousands of dollars). Sales were 13 thousand for x=4x = 4. Does the model fit this point, and what does the slope mean?

Example 17

medium
Two quantities satisfy: doubling xx multiplies yy by 8. Which power model y=xky = x^k fits, and what is kk?

Example 18

challenge
A 200 mg dose of medicine leaves the body so that 25% remains each hour, and a maintenance dose of 60 mg is added every hour. Write the recurrence, find the long-run equilibrium amount, and interpret it.

Example 19

challenge
A cylindrical can must hold 1000 cm3^3. Write its surface area (top, bottom, side) as a function of radius rr alone.

Example 20

challenge
Sales data: (0,5)(0, 5), (1,8)(1, 8), (2,13)(2, 13), (3,20)(3, 20) thousand. The second differences are constant. Find the quadratic model f(x)=ax2+bx+cf(x) = ax^2 + bx + c.

Example 21

medium
A pool contains 300 L and is filled at 25 L per minute. Write V(t)V(t) and find when it reaches 800 L.

Example 22

medium
A rumor model predicts the number who have heard it as N(t)=50β‹…3tN(t) = 50 \cdot 3^t (t in days). When does the count first exceed 1000? (Use that 34=813^4 = 81, 35=2433^5 = 243.)

Example 23

easy
A gym charges $25 to join plus $15 per month. Write the cost CC as a function of months mm.

Example 24

easy
A circle has radius rr. Write its area AA as a function of rr.

Example 25

easy
A culture starts with 50 cells and triples every hour. Write count NN after tt hours.

Example 26

easy
What is a sensible domain for a model of 'hours studied' as input?

Example 27

easy
A linear trend fits the data (1,3),(2,5),(3,7)(1,3), (2,5), (3,7). What is the linear function?

Example 28

medium
A ball is thrown so its height is h(t)=βˆ’4.9t2+19.6th(t) = -4.9t^2 + 19.6t meters. Find the time to reach maximum height.

Example 29

medium
A radioactive sample halves every 5 days. If 80 g remain at t=0t = 0, write M(t)M(t) in days.

Example 30

medium
A factory's cost is $500 setup plus $3 per unit. Write average cost per unit A(x)A(x) for xx units.

Example 31

medium
A loaf of bread cools so temperature is T(t)=25+75eβˆ’0.1tT(t) = 25 + 75 e^{-0.1 t} in Β°C, tt in minutes. Find the room temperature in the model.

Example 32

medium
A pendulum's period is approximately T(L)=2Ο€L/9.8T(L) = 2\pi \sqrt{L/9.8} seconds, LL in meters. Find TT when L=1L = 1 m.

Example 33

medium
Doubling the side of a cube multiplies the volume by what factor?

Example 34

medium
A bank account earns 4% annual interest compounded annually with $1000 initial. Write A(t)A(t) years later.

Example 35

medium
A model P(t)=200(1.05)tP(t) = 200 (1.05)^t fits population growth. What is the annual percent growth rate, and the population at t=10t = 10?

Example 36

hard
A box has square base side xx and height hh, volume V=x2h=4000V = x^2 h = 4000 cmΒ³. Write surface area S(x)S(x) (open top).

Example 37

hard
A drug has initial dose 200200 mg and the body removes 30%30\% per hour. Write M(t)M(t) and find when MM first drops below 5050 mg.

Example 38

hard
A square page has total area 200200 inΒ². Margins reduce the printable area: 22-in margins on top and bottom, 11-in margins on the sides. Express printable area P(x)P(x) as a function of page width xx (assuming square page).

Example 39

hard
A vehicle's fuel use is modeled F(v)=0.04v2βˆ’2v+35F(v) = 0.04 v^2 - 2 v + 35 L/h at speed vv km/h. Find the speed minimizing FF.

Example 40

hard
Pizza order: small (8 in) is \$8, large (16 in) is \$24. Using a model 'price proportional to area', is the large a better deal per square inch?

Example 41

hard
A model T(t)=A+Bsin⁑(Ο‰t+Ο•)T(t) = A + B \sin(\omega t + \phi) describes daily temperature. Which parameters control mean temperature, amplitude, and period?

Example 42

challenge
A lake has 10,000 fish; population follows logistic growth Pβ€²(t)=0.5P(1βˆ’P/10000)P'(t) = 0.5 P (1 - P/10000). What is the long-run population if started above 0?

Example 43

challenge
A pollutant in a lake decays exponentially with half-life 8 years. After 24 years, what fraction of the original amount remains?
← Back to Functional Modeling Practice Mode

Background Knowledge

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

function definitionmodeling with equations