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 โ€” functions, equations, distributions โ€” 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 NN 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 v0v_0 at angle ฮธ\theta, the range model (no air resistance) is R=v02sinโก(2ฮธ)gR = \frac{v_0^2 \sin(2\theta)}{g}. With v0=20v_0 = 20 m/s, ฮธ=30โˆ˜\theta = 30^\circ, g=10g = 10 m/s2^2, find RR.

Example 3

medium
A logistic population model is P(t)=K1+Aeโˆ’rtP(t) = \frac{K}{1 + Ae^{-rt}}. As tโ†’โˆžt \to \infty, what value does PP approach? What does this represent?

Example 4

medium
In a SIR epidemic model, SS, II, RR stand for what?

Example 5

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 6

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 7

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 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\$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.

Example 10

easy
A tree grows roughly 0.50.5 m per year. If it is 22 m today, model its height hh after tt years.

Example 11

medium
To compare two cell phone plans โ€” Plan A: $25+$0.05\$25 + \$0.05 per text; Plan B: $40+$0.02\$40 + \$0.02 per text โ€” model both costs as functions of tt 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 0โ‰คxโ‰ค500\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 14

easy
A pizza shop sells pp pizzas for $12\$12 each. Model the revenue RR as a function of pp.

Example 15

easy
For each 1-degree Celsius rise, ice cream sales rise by about 4040 cones. At 20โˆ˜20^\circC the shop sells 300300 cones. Model sales SS vs temperature TT for Tโ‰ฅ20T\ge 20.

Example 16

easy
A spring's force is modeled as F=โˆ’kxF=-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\$30 sign-up fee plus $15\$15 per month. Write a model for total cost CC after mm months.

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

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).
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