Practice Mathematical Modeling in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

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

Example 2

medium
For a projectile launched from ground level at speed v0 at angle θ, the range model (no air resistance) is R=v02sin⁡(2θ)g. With v0=20 m/s, θ=30∘, g=10 m/s2, find R.

Example 3

medium
A logistic population model is P(t)=K1+Ae−rt. As t→∞, what value does P approach? What does this represent?

Example 4

medium
In a SIR epidemic model, S, I, R stand for what?

Example 5

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

Example 6

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

Example 7

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

Example 8

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 9

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

Example 10

easy
A tree grows roughly 0.5 m per year. If it is 2 m today, model its height h after t years.

Example 11

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

Example 12

easy
Name the modeling assumption built into 'frictionless surface' in a physics problem.

Example 13

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

Example 14

easy
A pizza shop sells p pizzas for $12 each. Model the revenue R as a function of p.

Example 15

easy
For each 1-degree Celsius rise, ice cream sales rise by about 40 cones. At 20∘C the shop sells 300 cones. Model sales S vs temperature T for T≥20.

Example 16

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

Example 17

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

Example 18

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

Example 19

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

Example 20

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