Mathematical Modeling - ACT Math

A company sells widgets for $5 each. Which equation represents the total revenue $R$ from selling $n$ widgets?
Answer: $R = 5n$ This is a revenue calculation using a unit price model where R represents total revenue and n represents number of widgets sold. Each widget sells for $5, creating a direct proportional relationship between quantity and revenue. The equation R = 5n correctly represents total revenue as price per unit times number of units sold. There's no fixed revenue component (y-intercept is 0) since selling 0 widgets generates $0 revenue. Choices B and C incorrectly add a constant $5 revenue regardless of sales quantity.
A student earns money by tutoring. The relationship between hours tutored $x$ and total earnings $y$ (in dollars) is modeled by $y = 18x + 25$ . Based on the model $y = mx + b$ , what is the predicted value of $y$ when $x = 6$ ? Show the substitution: $y = 18(6) + 25$ .
Answer: 133 The linear model y = 18x + 25 shows earnings where y is total dollars and x is hours tutored. The slope 18 means the student earns $18 per hour, and the intercept 25 represents a base payment of $25. To find earnings after 6 hours, substitute x = 6: y = 18(6) + 25 = 108 + 25 = 133. Choice B shows the common error of forgetting to add the y-intercept, calculating only 18(6) = 108.
A plant's height increases by the same amount each week. It is 12 cm tall at week 0 and 27 cm tall at week 5. Let $x$ be the number of weeks and $y$ be the height in centimeters. Which equation best models the relationship between $x$ and $y$ ?
Answer: $y = 3x + 12$ This is a linear model where y represents plant height and x represents weeks. The plant grows from 12 cm to 27 cm over 5 weeks, so the slope is (27-12)/(5-0) = 15/5 = 3 cm per week. The y-intercept is 12 cm, which is the initial height at week 0. The equation is y = 3x + 12. Checking: when x = 0, y = 3(0) + 12 = 12 ✓; when x = 5, y = 3(5) + 12 = 27 ✓. Choice C reverses the slope and intercept values, while choice D uses incorrect coefficient values.
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?
Answer: The taxi charges $2.25 per mile. 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.
A water tank is being filled at a constant rate. It contains 50 liters at time $x=0$ minutes and 170 liters at time $x=6$ minutes. Let $x$ be time in minutes and $y$ be the amount of water in liters. Which equation best models the relationship between $x$ and $y$ ?
Answer: $y = 20x + 50$ This is a linear model where y represents water volume and x represents time in minutes. The tank contains 50 liters initially and 170 liters after 6 minutes, so the slope is (170-50)/(6-0) = 120/6 = 20 liters per minute, indicating water is added at 20 liters per minute. The y-intercept is 50 liters, which is the initial amount at time x = 0. The equation is y = 20x + 50. Checking: when x = 0, y = 20(0) + 50 = 50 ✓; when x = 6, y = 20(6) + 50 = 170 ✓. Choice B reverses the slope and intercept values.
A phone plan charges $25 per month plus $0.10 per text message. Which variable represents the number of text messages in the equation $y = 0.10x + 25$ ?
Answer: $x$ 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.
The equation $H = 180 - 10d$ models the height of a candle in centimeters, where $H$ is the height and $d$ is the number of days it has been burning. What is the height after 12 days?
Answer: 60 cm This candle height model H = 180 - 10d shows height H after d days of burning. The y-intercept 180 is initial height, and slope -10 means the candle burns down 10 cm per day. After 12 days: H = 180 - 10(12) = 180 - 120 = 60 cm. Choice C incorrectly adds instead of subtracts the burned portion.
A student saves money each week. The amount saved is modeled by $y = 15x + 5$ , where $x$ is the number of weeks and $y$ is the total amount saved in dollars. According to the model, how much money will the student have saved after $8$ weeks?
Answer: $125$ This is a linear model y = 15x + 5 where y represents total amount saved and x represents weeks. The slope 15 means the student saves $15 each week. The intercept 5 represents $5 already saved at the start (week 0). To find the total after 8 weeks, substitute x = 8: y = 15(8) + 5 = 120 + 5 = 125 dollars. The model predicts the student will have saved $125 total after 8 weeks. Choice A would result from forgetting the initial $5, while other choices involve calculation errors.
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 $y$ ?
Answer: $y = -2x + 20$ 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.
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?
Answer: $w = 100 + 10t$ 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.
A gym charges a one-time sign-up fee and a monthly membership fee. The total cost after $x$ months is modeled by $y = 25x + 40$ , where $y$ is the total cost in dollars. What is the meaning of the $y$ -intercept in this context?
Answer: The one-time sign-up fee is $40. This is a linear model $y = 25x + 40$ where y represents total cost and x represents months of membership. The slope $25$ means the monthly fee is $25 per month. The y-intercept $40$ represents the value of y when x = 0, which is the total cost when zero months have passed - this is the one-time sign-up fee of $40. When x = 0 months, $y = 25(0) + 40 = 40$ , confirming this initial cost. Choice A incorrectly describes the slope, while choice B confuses the slope with a monthly increase rather than the monthly rate.
The cost of producing an item is given by the equation $C = 100 + 5x$ , where $C$ is the cost in dollars and $x$ is the number of items produced. What is the cost for producing 20 items?
Answer: $500 This production cost model C = 100 + 5x shows cost C for x items produced. The y-intercept 100 represents fixed costs, and slope 5 means each additional item costs $5 to produce. For 20 items: C = 100 + 5(20) = 100 + 400 = $500. Choice A incorrectly uses only the variable cost portion.
A boat travels downstream and covers 40 miles in 2 hours. What is the boat's speed in miles per hour?
Answer: 20 mph This is a speed calculation problem using the relationship speed = distance ÷ time. The boat travels 40 miles in 2 hours downstream. To find speed: 40 miles ÷ 2 hours = 20 miles per hour. This represents the boat's actual travel speed under these conditions. The calculation is straightforward division of total distance by total time. The other choices result from incorrect arithmetic or misunderstanding the speed formula.
A delivery service charges according to the linear model $y = 3.5x + 6$ , where $x$ is the number of miles driven and $y$ is the total charge in dollars. Based on the model, what is the predicted charge when $x = 10$ miles? (Substitute into $y = 3.5x + 6$ .)
Answer: $41$ This is a linear model y = 3.5x + 6 where y represents total charge and x represents miles driven. The slope 3.5 means the charge increases $3.50 per mile driven. The intercept 6 represents a $6 base charge when no miles are driven. To find the charge when x = 10 miles, substitute into the equation: y = 3.5(10) + 6 = 35 + 6 = 41 dollars. The model predicts a $41 total charge for a 10-mile delivery. Choice B would result from forgetting to add the base charge of $6.
A runner's distance from the starting line increases at a constant rate. The relationship is modeled by $y = 0.25x$ , where $x$ is time in seconds and $y$ is distance in meters. What is the meaning of the slope in this context?
Answer: The runner runs 0.25 meters per second. 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.
The temperature on a winter day is modeled by the equation $T = -3h + 12$ , where $T$ is the temperature in degrees Celsius and $h$ is the number of hours since midnight. What is the temperature at 5 AM?
Answer: -3°C This temperature model T = -3h + 12 shows temperature T at hour h since midnight. The y-intercept 12 is temperature at midnight, and slope -3 means temperature drops 3°C per hour. At 5 AM (h = 5): T = -3(5) + 12 = -15 + 12 = -3°C. Choice B gives the positive value, ignoring the negative slope.
A taxi company charges a flat rate of $2 plus $1.25 per mile driven. Which equation represents the total cost $C$ for a trip of $m$ miles?
Answer: $C = 2 + 1.25m$ This is a linear cost model, where the flat rate is the constant and the per-mile rate is the coefficient of the distance. The $2 flat rate is charged once for the trip, and the $1.25 applies to each of the $m$ miles, so the mileage cost is $1.25m$, giving $C = 2 + 1.25m$ . Both $C = 1.25 + 2m$ and $C = 2m + 1.25$ swap the roles, charging $2 per mile with a $1.25 flat rate. $C = 2m - 1.25$ repeats that swap and subtracts, which would shrink the fare as the trip lengthens. Attach the amount whose description includes the word "per" to the variable, and let the amount charged once stand alone as the constant.
The cost of attending a concert is modeled by the equation $C = 25n + 5$ , where $C$ is the total cost in dollars and $n$ is the number of tickets. What is the cost for 3 tickets?
Answer: $80 This concert cost model C = 25n + 5 shows total cost C for n tickets. The slope 25 means each ticket costs $25, and y-intercept 5 represents a flat service fee. For 3 tickets: C = 25(3) + 5 = 75 + 5 = $80. Choice B incorrectly uses the per-ticket cost without the service fee.
A train travels at a constant speed of 80 kilometers per hour. How far will it travel in 3.5 hours?
Answer: 280 kilometers 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.
The sales of a product are predicted by the equation $S = 200 + 15d$ , where $S$ is the number of units sold and $d$ is the number of days. What is the predicted sales after 10 days?
Answer: 350 units This sales prediction model S = 200 + 15d shows units sold S after d days. The y-intercept 200 represents initial or baseline sales, and slope 15 means sales increase by 15 units per day. After 10 days: S = 200 + 15(10) = 200 + 150 = 350 units. Choice B incorrectly multiplies by 100 instead of adding.

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