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ACT Math Help: Mathematical Modeling

Review real example questions for Mathematical Modeling in ACT Math.

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A taxi fare is modeled by $y = 2.25x + 4.50$ , where $x$ is the number of miles traveled and $y$ is the total fare in dollars.

What does the slope represent in this context?

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Question 1

A taxi fare is modeled by $y = 2.25x + 4.50$ , where $x$ is the number of miles traveled and $y$ is the total fare in dollars.

What does the slope represent in this context?

The taxi charges a $4.50 starting fee.
The taxi charges $4.50 per mile.
The taxi charges a $2.25 starting fee.
The taxi charges $2.25 per mile. (correct answer)

Explanation: In a linear model $y = mx + b$ , the slope $m$ is the amount $y$ changes for each one-unit increase in $x$ , while $b$ is the value of $y$ when $x = 0$ . In $y = 2.25x + 4.50$ the slope is $2.25$ and it multiplies the miles, so it means the taxi charges $2.25 per mile; the $4.50 is what you owe before traveling any miles. Saying the taxi charges a $4.50 starting fee is a true statement about the model but describes the intercept, not the slope, which is exactly the trap; saying $4.50 per mile attaches the intercept's value to the slope's meaning, and saying $2.25 is a starting fee does the reverse swap. Identify which number is multiplied by the variable and which stands alone: the multiplied one is always the per-unit rate, and the standalone one is always the fixed starting amount.

Question 2

The temperature $y$ (in °C) in a freezer changes linearly with time $x$ (in minutes). It is $20$ °C at $x=0$ and $-4$ °C at $x=12$ .

Which equation best models the relationship between $x$ and $y$ ?

$y = -12x + 4$
$y = 2x + 20$
$y = -4x + 12$
$y = -2x + 20$ (correct answer)

Explanation: A quantity changing linearly with time is modeled by $y=mx+b$ , where $b$ is the value at $x=0$ and $m$ is the change per unit of time. The temperature is $20$ at $x=0$ , so the constant is $20$ , and the slope is $\dfrac{-4-20}{12-0}=\dfrac{-24}{12}=-2$ degrees per minute, giving $y=-2x+20$ ; checking, $x=12$ produces $-2(12)+20=-4$ as required. The equation $y=2x+20$ has the correct starting value but a positive rate, which would warm the freezer instead of cooling it. The equation $y=-4x+12$ mistakenly uses the given data values themselves as slope and intercept, and $y=-12x+4$ does the same thing with the roles further scrambled. Identify the value at time zero as the constant first, then compute the rate as change in output over change in input, and confirm the model by testing the second data point.

Question 3

A ferry service charges $15 per passenger plus a flat fee of $100 per trip. Which equation represents the total cost $T $for$ p$ passengers?

$T = 15p + 100$ (correct answer)
$T = 100p + 15$
$T = 15 + 100p$
$T = 100 - 15p$

Explanation: This is a linear cost model, where a per-unit rate multiplies the variable and a one-time fee stays constant. The $15 charge applies once for each passenger, so it becomes $15p$ , while the $100 flat fee is charged once per trip no matter how many passengers ride, so it stays as $+100 $, giving$ T = 15p + 100 $. Both$ T = 100p + 15 $and$ T = 15 + 100p $swap those two roles, charging $100 for every passenger and treating the $15 as the one-time trip fee. $T = 100 - 15p$ makes every extra passenger lower the total, which turns a charge into a discount. Decide which quantity repeats with the variable and which happens only once: the repeating amount gets the variable attached, and the one-time amount is the constant.

Question 4

A storage tank is being filled with water at a constant rate of 10 gallons per minute. If the tank starts at 100 gallons, what is the equation for the amount of water $w$ in the tank after $t$ minutes?

$w = 100 - 10t$
$w = 10 + 100t$
$w = 100 + 10t$ (correct answer)
$w = 100t + 10$

Explanation: This is a linear accumulation model, where the starting amount is the constant and the filling rate multiplies the time. At $t = 0$ the tank already holds 100 gallons, so 100 is the constant, and each minute adds 10 more gallons, so the accumulated water is $10t$ , giving $w = 100 + 10t$ . Both $w = 10 + 100t$ and $w = 100t + 10$ reverse the roles, starting the tank at 10 gallons and filling it at 100 gallons per minute. $w = 100 - 10t$ has the right numbers in the right places but subtracts, which would model a tank draining rather than being filled. Check the sign against the story first: filling and growing mean addition, while draining and shrinking mean subtraction.

Question 5

A cell phone plan costs $30 per month plus $0.15 per text message. Which equation represents the total monthly cost $C $for sending$ t$ text messages?

$C = 30.15t$
$C = 0.15 + 30t$
$C = 30t + 0.15$
$C = 30 + 0.15t$ (correct answer)

Explanation: This is a linear cost model, where the fixed monthly charge is the constant and the per-text rate multiplies the number of texts. The $30 is paid once each month regardless of texting, and each of the $t $messages costs $0.15, so the message cost is$ 0.15t $, giving$ C = 30 + 0.15t $. The expression$ C = 30.15t $adds the two amounts together and then multiplies everything by$ t $, which wrongly charges the $30 monthly fee once per text message. Both $C = 0.15 + 30t$ and $C = 30t + 0.15$ swap the roles, making $30 the per-text rate and $0.15 the monthly fee. Only the amount that repeats with each unit belongs next to the variable, so never fold a one-time fee into the coefficient.

Question 6

A taxi company charges a base fare of $3 plus $2 per mile. Which equation best models the relationship between the total fare $y $and the number of miles$ x$ traveled?

$y = 2 + 3x$
$y = 3x + 2$
$y = 3 - 2x$
$y = 2x + 3$ (correct answer)

Explanation: This is a linear cost model in which the base fare is the constant and the per-mile rate multiplies the number of miles. The $2 charge repeats for every mile, so it becomes $2x$ , and the $3 base fare is paid once regardless of distance, so it is the constant, giving $y = 2x + 3 $. Both$ y = 2 + 3x $and$ y = 3x + 2 $reverse those roles, charging $3 per mile with a $2 base fare. $y = 3 - 2x$ makes the fare shrink as the trip gets longer, which contradicts a charge that accumulates. Ask what happens at $x = 0$ : the correct model must return the base fare alone, which is a fast way to confirm the constant term.

Question 7

A plant grows at a constant rate of 2 cm per day. If the plant is initially 5 cm tall, which equation models the height $h$ of the plant after $d$ days?

$h = 5 + 2d$ (correct answer)
$h = 2 + 5d$
$h = 5d + 2$
$h = 5 - 2d$

Explanation: This is a linear growth model, where the initial height is the constant and the growth rate multiplies the number of days. The plant starts at 5 cm, so 5 is the constant, and it gains 2 cm each day, so the growth after $d$ days is $2d$ , giving $h = 5 + 2d$ . Both $h = 2 + 5d$ and $h = 5d + 2$ swap the two numbers, starting the plant at 2 cm and growing it 5 cm per day. $h = 5 - 2d$ uses the right numbers in the right roles but subtracts, describing a plant that shrinks 2 cm per day. Test the model at time zero: whatever remains when the variable is 0 must be the starting value.

Question 8

A gym charges a $50 monthly fee and $10 per class attended. Which equation represents the total monthly cost $T $for attending$ c$ classes?

$T = 50c + 10$
$T = 10 + 50c$
$T = 50 + 10c$ (correct answer)
$T = 50 - 10c$

Explanation: This is a linear cost model where the fixed monthly fee is the constant and the per-class rate multiplies the number of classes. The $50 is charged once a month whether or not any classes are attended, and each class costs $10, so the class cost is $10c$ , giving $T = 50 + 10c$ . Both $T = 50c + 10$ and $T = 10 + 50c$ reverse the roles, charging $50 for every class and treating the $10 as the monthly fee. $T = 50 - 10c$ keeps the numbers in the right roles but subtracts, so attending classes would reduce the bill instead of raising it. Confirm the direction of change before choosing: if the total should rise as the variable grows, the rate must be added, not subtracted.

Question 9

A phone plan charges $25 per month plus $0.10 per text message. Which variable represents the number of text messages in the equation $y = 0.10x + 25$?

0.10
$y$
$x$ (correct answer)
25

Explanation: In the phone plan equation y = 0.10x + 25, we need to identify what each variable represents based on the context. The equation models total monthly cost where y represents the total cost, 25 represents the fixed monthly fee, and 0.10 represents the cost per text message. Therefore, x must represent the number of text messages sent, since it's the variable being multiplied by the per-text rate. The structure follows the pattern: total cost = (rate per text)(number of texts) + fixed fee.

Question 10

A hot air balloon is descending at a rate of 5 meters per minute. If its initial altitude is 200 meters, what equation models the altitude $y$ after $x$ minutes?

$y = 200 - x$
$y = 5x + 200$
$y = 200 + 5x$
$y = 200 - 5x$ (correct answer)

Explanation: This is a linear altitude model where y represents altitude and x represents time in minutes. The slope of -5 means the altitude decreases by 5 meters per minute (negative because it's descending). The y-intercept of 200 represents the initial altitude when x = 0 minutes. The equation y = 200 - 5x correctly models this decreasing relationship. Choice B incorrectly uses a positive slope, which would represent ascending rather than descending. Choice A uses the wrong rate of change.