Practice Functional Modeling in Math

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

Worksheet PDF Free Answer-key PDF Family

Quick 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.

Showing a random 20 of 50 problems.

Example 1

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 2

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

Example 3

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 4

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

Example 5

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

Example 6

hard
A cylindrical can must hold 500500 cm³. Express the total surface area SS as a function of radius rr, and find the value of rr that minimizes material use.

Example 7

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 8

challenge
A pollutant in a lake decays exponentially with half-life 8 years. After 24 years, what fraction of the original amount remains?

Example 9

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

Example 10

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 11

hard
A square page has total area 200200 in². 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 12

easy
A truck travels at constant 60 mph. Write distance dd as a function of hours tt.

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 ball is thrown up; its height is h(t)=−5t2+20th(t) = -5t^2 + 20t meters. Find when it lands (height returns to 0) and state why the model is invalid afterward.

Example 15

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

Example 16

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 17

medium
A loaf of bread cools so temperature is T(t)=25+75e−0.1tT(t) = 25 + 75 e^{-0.1 t} in °C, tt in minutes. Find the room temperature in the model.

Example 18

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

Example 19

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

Example 20

easy
A circle has radius rr. Write its area AA as a function of rr.
← Back to Functional Modeling Worked Examples