Mathematical Modeling Examples in Math

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

The process of using mathematical structures — functions, equations, distributions — to represent, analyze, and predict real-world phenomena.

Building a mathematical version of reality to understand and predict.

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: Mathematical modeling turns a real situation into a function or equation so you can compute and forecast answers you could not just look up.

Common stuck point: The procedure for mathematical modeling is the easy part; the trap is picking the model that is easiest to compute instead of the one matching the situation. Asking "Am I being asked to invent the relationship between real-world quantities, not just compute with one already given?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I being asked to invent the relationship between real-world quantities, not just compute with one already given?

Worked Examples

Example 1

easy
A taxi charges a base fare of $2.50\$2.50 and $1.20\$1.20 per kilometre. Write a mathematical model for the total fare FF as a function of distance dd (km), identify variables, and compute the fare for a 7 km ride.

Answer

F(d)=2.50+1.20d,F(7)=$10.90F(d) = 2.50 + 1.20d,\quad F(7) = \$10.90

First step

1
Identify variables: FF = total fare ($), dd = distance (km).

Full solution

  1. 2
    Model: F(d)=2.50+1.20dF(d) = 2.50 + 1.20d.
  2. 3
    For d=7d = 7: F(7)=2.50+1.20×7=2.50+8.40=10.90F(7) = 2.50 + 1.20 \times 7 = 2.50 + 8.40 = 10.90.
A mathematical model translates a real-world situation into equations. Identifying what changes (variables) and what is fixed (parameters) is the first step in building any model.

Example 2

medium
A population of bacteria doubles every hour. If the initial count is P0=500P_0 = 500, write a model for the population P(t)P(t) after tt hours and find P(4)P(4).

Example 3

medium
A pool fills at 55 L/min for the first 2020 min, then 88 L/min after that. Build a piecewise model for the volume V(t)V(t) (in liters) for 0≤t≤600\le t\le 60 minutes.

Example 4

medium
A box has square base of side xx cm and height hh cm with volume 10001000 cm3^3. Express surface area AA as a function of xx only.

Example 5

medium
To compare two cell phone plans — Plan A: $25+$0.05\$25 + \$0.05 per text; Plan B: $40+$0.02\$40 + \$0.02 per text — model both costs as functions of tt texts and find the break-even point.

Example 6

medium
A logistic population model is P(t)=K1+Ae−rtP(t) = \frac{K}{1 + Ae^{-rt}}. As t→∞t \to \infty, what value does PP approach? What does this represent?

Example 7

hard
A drug's concentration in the blood is modeled as C(t)=C0e−ktC(t) = C_0 e^{-kt}. If the half-life is 44 hours, find kk.

Example 8

hard
A bank account compounds continuously at 5%5\% annual rate. Model the balance B(t)B(t) with initial deposit $1000\$1000, and find when it doubles.

Example 9

challenge
To model how long until two people in a room of nn share a birthday, derive the probability that at least two share a birthday and find the smallest nn for which this exceeds 50%50\%.

Practice Problems

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

Example 1

easy
A rectangle has perimeter PP and length ll. Express the width ww as a function of PP and ll, then find ww when P=30P=30 and l=8l=8.

Example 2

medium
A car travels at a constant speed vv km/h. Write a model for distance dd after tt hours. If the car must reach a destination 240 km away in 3 hours, what speed is required?

Example 3

easy
A problem describes a savings account growing by a fixed percent each year. Which mathematical structure models this?

Example 4

easy
A car travels at constant speed. Which function models distance versus time?

Example 5

easy
A population doubles every hour. What kind of model fits?

Example 6

easy
A modeler writes 'assume the ball is a point mass.' Is this an assumption of the model or a prediction of it?

Example 7

easy
A model fit to data for 0≤x≤100 \le x \le 10 is used to predict at x=1000x=1000. What modeling error is this?

Example 8

easy
True or false: a good model exactly replicates reality.

Example 9

easy
To decide how many buses are needed for 130130 students with 4040 seats each, which model fits: division then rounding up, or plain division?

Example 10

easy
A spring's force is modeled as F=−kxF=-kx. The negative sign encodes which real feature?

Example 11

medium
A disease spreads slowly at first, then rapidly, then levels off as people recover or are immune. Which model captures all three phases?

Example 12

medium
A modeler assumes population growth is linear, but data show it doubling each period. What modeling mistake was made?

Example 13

medium
A projectile model gives a maximum height of −5-5 meters for some input. What does this reveal about the model or its inputs?

Example 14

medium
To model the time to empty a draining tank, you use V(t)=V0−rtV(t)=V_0 - rt. What does this model assume about the drain rate?

Example 15

medium
Two models fit data equally well, but one uses 22 parameters and one uses 99. Which is generally preferred and why?

Example 16

medium
A model predicts 40%40\% chance of rain. A critic says 'it rained, so the model was wrong.' Why is the critic mistaken about probabilistic models?

Example 17

medium
A linear model for ice cream sales vs temperature predicts negative sales at −20∘-20^\circC. What modeling principle does this violate?

Example 18

medium
Modeling traffic, you treat individual cars as a continuous 'fluid' density. What is gained and what is the key assumption?

Example 19

medium
A model of a falling object near Earth uses constant acceleration gg. What real feature does fixing gg ignore, and when does it matter?

Example 20

challenge
You model bacteria with unbounded exponential growth P=P0ertP=P_0 e^{rt}. Explain when this model breaks down and what feature a better model must add.

Example 21

challenge
A coin-flip game pays $2n\$2^n if the first head appears on flip nn. The expected-value model gives infinite value, yet no one pays much to play. What does this reveal about the model's assumptions?

Example 22

challenge
To model the spread of a rumor in a school of NN students, derive why the rate is proportional to (knowers)(non-knowers) and identify the resulting model.

Example 23

easy
A gym charges a $30\$30 sign-up fee plus $15\$15 per month. Write a model for total cost CC after mm months.

Example 24

easy
A radioactive sample loses half its mass every 55 years. Which model fits: linear decay, exponential decay, or logistic?

Example 25

easy
The temperature of a cup of coffee approaches room temperature over time. Which simple model shape captures this?

Example 26

easy
For each 1-degree Celsius rise, ice cream sales rise by about 4040 cones. At 20∘20^\circC the shop sells 300300 cones. Model sales SS vs temperature TT for T≥20T\ge 20.

Example 27

medium
A bacteria culture grows from 200200 to 18001800 in 33 hours. Model the population as P(t)=P0ertP(t) = P_0 e^{rt} and find rr.

Example 28

medium
A linear model predicts test score S=5h+50S = 5h + 50 where hh is study hours. The model would predict S=200S = 200 at h=30h = 30. What modeling error is this?

Example 29

medium
A phone plan charges $20/month\$20/month plus $0.10\$0.10 per minute over 500500 minutes. Model the monthly bill BB for mm minutes used.

Example 30

medium
For a projectile launched from ground level at speed v0v_0 at angle θ\theta, the range model (no air resistance) is R=v02sin⁡(2θ)gR = \frac{v_0^2 \sin(2\theta)}{g}. With v0=20v_0 = 20 m/s, θ=30∘\theta = 30^\circ, g=10g = 10 m/s2^2, find RR.

Example 31

medium
A model is calibrated using data from 0≤x≤500\le x\le 50. The model's R2=0.97R^2 = 0.97 on that range. Why is it still risky to use the model at x=200x = 200?

Example 32

medium
A car's stopping distance is often modeled as d=v+v220d = v + \frac{v^2}{20} (with dd in meters and vv in m/s). What real factors does the v2/20v^2/20 term encode?

Example 33

medium
A model gives the time to fill a tank as T=VrT = \frac{V}{r} where VV is volume and rr is flow rate. What modeling assumption hides in this formula?

Example 34

hard
You have 2424 meters of fencing to enclose a rectangular garden against a wall (no fence needed on the wall side). Model the enclosed area AA as a function of one side and find the maximum.

Example 35

hard
A predator-prey (Lotka-Volterra) model gives oscillating populations RR (rabbits) and FF (foxes). What real feature does the oscillation capture, and what assumption keeps oscillations from dying out?

Example 36

hard
You fit a parabola y=ax2+bx+cy = ax^2 + bx + c to three data points exactly. Why is this not necessarily a better model than a line that fits them approximately?

Example 37

challenge
A sandbox model: every grain of sand dropped on a pile either stays or causes an avalanche of various sizes. A simple model predicts a power-law distribution of avalanche sizes. What property of the model lets it produce avalanches of every scale?
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Background Knowledge

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

abstraction