Home

Tutoring

Subjects

Live Classes

Study Coach

Essay Review

On-Demand Courses

Colleges

Games

Opening subject page...

Loading your content

ACT Math

ACT Math Help: Mathematical Modeling

Review real example questions for Mathematical Modeling in ACT Math.

Question 1

A savings account earns simple interest according to the equation A=1000+50tA = 1000 + 50tA=1000+50t, where AAA is the amount in dollars and ttt is the number of years. How much interest is earned after 5 years?

  1. $1250
  2. $1000
  3. $250
  4. $1250
Explanation: This simple interest model A = 1000 + 50t shows account balance A where t is years. The y-intercept 1000 is the initial principal, and slope 50 is the annual interest earned. After 5 years, total balance is A = 1000 + 50(5) = 1250. The interest earned is the difference: 1250 - 1000 = $250\$250$250. Choice A confuses total balance with interest earned.

Question 2

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 hhh of the plant after ddd days?

  1. h=5+2dh = 5 + 2dh=5+2d
  2. h=2+5dh = 2 + 5dh=2+5d
  3. h=5d+2h = 5d + 2h=5d+2
  4. h=2d+5h = 2d + 5h=2d+5
Explanation: This is a linear growth model where h represents height and d represents days. The slope of 2 means the plant grows 2 cm per day. The y-intercept of 5 represents the initial height when d = 0 days. The equation h = 5 + 2d correctly models this relationship where current height equals initial height plus growth over time. Choice D incorrectly reverses the coefficients, making the initial height 2 cm and growth rate 5 cm per day.

Question 3

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

  1. 0.10
  2. yyy
  3. xxx
  4. 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 4

A train travels at a constant speed of 80 kilometers per hour. How far will it travel in 3.5 hours?

  1. 280 kilometers
  2. 260 kilometers
  3. 300 kilometers
  4. 320 kilometers
Explanation: This is a distance calculation using constant speed where distance = speed × time. The train travels at 80 kilometers per hour for 3.5 hours. Using the relationship: distance = speed × time, we get 80 km/hour × 3.5 hours = 280 kilometers. This represents the total distance covered when traveling at constant speed for the given time duration. The calculation involves multiplying the rate by the time period.

Question 5

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 yyy after xxx minutes?

  1. y=200−xy = 200 - xy=200−x
  2. y=5x+200y = 5x + 200y=5x+200
  3. y=200+5xy = 200 + 5xy=200+5x
  4. y=200−5xy = 200 - 5xy=200−5x
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.

Question 6

The cost of manufacturing a product is given by the equation C=500+3xC = 500 + 3xC=500+3x, where CCC is the total cost in dollars and xxx is the number of units produced. What is the fixed cost in this context?

  1. $0
  2. $3
  3. $503
  4. $500
Explanation: This cost model C=500+3xC = 500 + 3xC=500+3x represents manufacturing costs where CCC is total cost and xxx is units produced. The y-intercept 500500500 represents the fixed cost - costs that remain constant regardless of production level (like rent, equipment, insurance). The slope 333 represents variable cost per unit. Choice B confuses the variable cost with fixed cost.

Question 7

A plant is 6 inches tall when it is purchased and grows at a constant rate of 1.5 inches per week. Let xxx be the number of weeks since purchase and let yyy be the plant's height (in inches). Which equation best models the relationship?

  1. y=6x+1.5y = 6x + 1.5y=6x+1.5
  2. y=1.5x+6y = 1.5x + 6y=1.5x+6
  3. y=6−1.5xy = 6 - 1.5xy=6−1.5x
  4. y=1.5−6xy = 1.5 - 6xy=1.5−6x
Explanation: This is a linear growth model where y is plant height in inches and x is weeks since purchase. The plant starts at 6 inches (when x = 0), making 6 the y-intercept. The plant grows 1.5 inches per week, making 1.5 the slope. The equation is y = 1.5x + 6. Choice A incorrectly reverses the slope and intercept, putting 6 as the growth rate and 1.5 as the starting height. Choices C and D use negative slopes, which would mean the plant is shrinking.

Question 8

A candle burns down at a constant rate. It is 18 cm tall at time x=0x=0x=0 hours and 12 cm tall at time x=3x=3x=3 hours. If xxx is time (hours) and yyy is height (cm), which equation best models the candle's height over time?

  1. y=2x+18y = 2x + 18y=2x+18
  2. y=−2x+18y = -2x + 18y=−2x+18
  3. y=−6x+18y = -6x + 18y=−6x+18
  4. y=2x−18y = 2x - 18y=2x−18
Explanation: This models a candle burning at constant rate where y is height in cm and x is time in hours. At x = 0, the candle is 18 cm tall (y-intercept = 18). At x = 3, it's 12 cm tall. The candle lost 6 cm in 3 hours, so the rate is -2 cm per hour (slope = -2). The equation is y = -2x + 18. Choice A uses positive slope, meaning the candle would grow taller. Choice C has slope -6, which would mean the candle loses 6 cm per hour instead of per 3 hours.

Question 9

A runner's distance from the starting line increases at a constant rate. The relationship is modeled by y=0.25xy = 0.25xy=0.25x, where xxx is time in seconds and yyy is distance in meters. What is the meaning of the slope in this context?

  1. The runner starts 0.25 meters ahead of the starting line.
  2. The runner runs 0.25 meters per second.
  3. The runner runs 4 seconds per meter.
  4. The runner runs 0.25 meters in total.
Explanation: The model y = 0.25x represents distance (y in meters) versus time (x in seconds) for a runner. The slope 0.25 means the runner's distance increases by 0.25 meters for each second that passes - this is the runner's speed of 0.25 meters per second. There is no y-intercept term, meaning the runner starts at the starting line (0 meters when x = 0). Choice C incorrectly inverts the units to seconds per meter. Choice A misinterprets the slope as a starting position.

Question 10

A gym charges a one-time sign-up fee and then a monthly fee. The total cost after xxx months is y=30x+80y = 30x + 80y=30x+80, where xxx is months and yyy is total cost (dollars).

What is the meaning of the slope in this context?

  1. The gym charges \80$ each month.
  2. The gym charges a sign-up fee of \30$.
  3. The gym charges \30$ per month.
  4. The gym charges a sign-up fee of \80$ each month.
Explanation: In the linear model y = 30x + 80 for gym costs, y is total cost and x is months. The slope (coefficient of x) is 30, which represents the rate of change - the cost increases by $30 for each additional month. This is the monthly fee. The y-intercept 80 is the initial cost when x = 0, representing the one-time sign-up fee. Choice B incorrectly identifies the slope as the sign-up fee instead of recognizing it as the monthly rate.