Functional Modeling Examples: 46 Problems with Answers

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: 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

40

m. Express the area

A

as a function of the width

w

, find the domain, and determine the width that maximizes area.

Answer

A(w)=20w-w^2

; domain

(0,20)

; maximum

100\text{ m}^2

w=10

First step

Perimeter:

2w+2l=40 \Rightarrow l=20-w

. Area:

A(w)=w\cdot l=w(20-w)=20w-w^2

Full solution

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

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.

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

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.

Example 3

easy

A taxi charges a $3 flat fee plus $2 per mile. Write the cost

C

as a function of miles

m

Example 4

easy

A bacteria count doubles every hour, starting at 100. Write the count

N

as a function of hours

t

Example 5

easy

A rectangle has a fixed perimeter of 20. Write its area

A

as a function of width

w

Example 6

easy

A phone plan costs $40 per month with no per-minute charge. Write monthly cost

C

as a function of minutes

m

Example 7

easy

Water drains from a 50-liter tank at 5 liters per minute. Write the volume

V

as a function of minutes

t

Example 8

easy

A function

f(x) = 2x

models dollars earned per hour worked,

x

. What is a sensible domain restriction for this real-world model?

Example 9

easy

A square's area as a function of side length

s

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) = -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) = P_0 b^t

Example 13

medium

A company's profit is revenue minus cost: revenue

R(x) = 10x

and cost

C(x) = 4x + 120

for

x

units. Write profit

P(x)

and find the break-even point.

Example 14

medium

A car depreciates 20% per year from $30000. Write its value

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

x

perpendicular to the wall.

Example 16

medium

A model predicts

f(x) = 3x + 1

thousand sales for ad spend

x

(thousands of dollars). Sales were 13 thousand for

x = 4

. Does the model fit this point, and what does the slope mean?

Example 17

medium

Two quantities satisfy: doubling

x

multiplies

y

by 8. Which power model

y = x^k

fits, and what is

k

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 cm

^3

. Write its surface area (top, bottom, side) as a function of radius

r

alone.

Example 20

challenge

Sales data:

(0, 5)

(1, 8)

(2, 13)

(3, 20)

thousand. The second differences are constant. Find the quadratic model

f(x) = ax^2 + bx + c

Example 21

medium

A pool contains 300 L and is filled at 25 L per minute. Write

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) = 50 \cdot 3^t

(t in days). When does the count first exceed 1000? (Use that

3^4 = 81

3^5 = 243

Example 23

easy

A gym charges $25 to join plus $15 per month. Write the cost

C

as a function of months

m

Example 24

easy

A circle has radius

r

. Write its area

A

as a function of

r

Example 25

easy

A culture starts with 50 cells and triples every hour. Write count

N

after

t

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)

. What is the linear function?

Example 28

medium

A ball is thrown so its height is

h(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 = 0

, write

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)

for

x

units.

Example 31

medium

A loaf of bread cools so temperature is

T(t) = 25 + 75 e^{-0.1 t}

in °C,

t

in minutes. Find the room temperature in the model.

Example 32

medium

A pendulum's period is approximately

T(L) = 2\pi \sqrt{L/9.8}

seconds,

L

in meters. Find

T

when

L = 1

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)

years later.

Example 35

medium

A model

P(t) = 200 (1.05)^t

fits population growth. What is the annual percent growth rate, and the population at

t = 10

Example 36

hard

A box has square base side

x

and height

h

, volume

V = x^2 h = 4000

cm³. Write surface area

S(x)

(open top).

Example 37

hard

A drug has initial dose

200

mg and the body removes

30\%

per hour. Write

M(t)

and find when

M

first drops below

50

mg.

Example 38

hard

A square page has total area

200

in². Margins reduce the printable area:

2

-in margins on top and bottom,

1

-in margins on the sides. Express printable area

P(x)

as a function of page width

x

(assuming square page).

Example 39

hard

A vehicle's fuel use is modeled

F(v) = 0.04 v^2 - 2 v + 35

L/h at speed

v

km/h. Find the speed minimizing

F

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 + 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

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?

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Related Concepts

Function Modeling with Equations Optimization Prediction

Background Knowledge

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

function definitionmodeling with equations