Math 1 Quiz: Interpreting Slope And Intercepts
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Interpreting Slope And InterceptsQuestion 1 of 20

A water tank is being filled at a constant rate. The relationship between time (in hours) and the volume of water (in gallons) is given by the equation V = 25t + 150, where V represents the volume and t represents time.

In this context, what does the y-intercept represent, and what constraint does it suggest about the tank?

The tank initially contained 150 gallons, suggesting it was partially filled before the filling process began
The tank can hold a maximum of 150 gallons, suggesting this is the tank's total capacity limit
Water is being added at 150 gallons per hour, suggesting this is the constant filling rate
The tank will be empty after 150 hours, suggesting this is when the draining process completes
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Math 1 Quiz

Math 1 Quiz: Interpreting Slope And Intercepts

Practice Interpreting Slope And Intercepts in Math 1 with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Interpreting Slope And Intercepts, giving you a quick way to practice the rules, question types, and explanations that matter most for Math 1.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

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Question 1

A water tank is being filled at a constant rate. The relationship between time (in hours) and the volume of water (in gallons) is given by the equation V = 25t + 150, where V represents the volume and t represents time.

In this context, what does the y-intercept represent, and what constraint does it suggest about the tank?

  1. The tank initially contained 150 gallons, suggesting it was partially filled before the filling process began (correct answer)
  2. The tank can hold a maximum of 150 gallons, suggesting this is the tank's total capacity limit
  3. Water is being added at 150 gallons per hour, suggesting this is the constant filling rate
  4. The tank will be empty after 150 hours, suggesting this is when the draining process completes
Explanation: The y-intercept occurs when t = 0, giving V = 25(0) + 150 = 150. This means at time zero (before filling begins), there were already 150 gallons in the tank. This suggests the tank was not empty initially. Choice B confuses the y-intercept with capacity, Choice C confuses it with the slope (rate), and Choice D misinterprets both the meaning and the filling vs. draining context.

Question 2

The height of a projectile follows H = -16t + 80, where H is height in feet and t is time in seconds after reaching maximum height.

What do the slope and y-intercept indicate about the projectile's motion and the reference point for this measurement?

  1. Rises 16 feet per second to maximum height of 80 feet, measuring from ground upward
  2. Falls 16 feet per second from maximum height of 80 feet, measuring from the peak downward (correct answer)
  3. Accelerates at 80 feet per second squared with initial velocity of 16 feet per second
  4. Travels horizontally 80 feet in 16 seconds, measuring total displacement from launch point
Explanation: When analyzing linear equations that model real-world motion, you need to interpret both the slope and y-intercept in the context of what's being measured. The equation H=−16t+80H = -16t + 80 is in slope-intercept form, where the slope is -16 and the y-intercept is 80. The negative slope of -16 tells you the projectile is falling at a rate of 16 feet per second. Since time is measured "after reaching maximum height," the projectile can only be moving downward from that point. The y-intercept of 80 represents the height when t=0t = 0, which is the moment the projectile reaches its peak—so the maximum height is 80 feet above the reference point. Choice B correctly captures both elements: the projectile falls 16 feet per second from its maximum height of 80 feet. Choice A misinterprets the negative slope as positive motion—a projectile cannot rise after reaching maximum height. Choice C confuses the linear velocity model with acceleration; this equation shows constant velocity, not acceleration, and neither coefficient represents acceleration or initial velocity in this context. Choice D completely misunderstands the equation by treating it as horizontal displacement rather than vertical height. Remember that the sign of the slope always indicates direction of change. In projectile motion problems, pay careful attention to what the time reference point represents—here it's "after maximum height," which immediately tells you the projectile must be falling, making a negative slope essential.

Question 3

A candle burns according to the equation L = -0.5h + 8, where L is length in inches and h is hours of burning.

What does this model indicate about the candle's physical properties and burning limitations?

  1. The candle burns at 8 inches per 0.5 hours and was originally 16 inches, lasting exactly 2 hours total
  2. The candle burns 8 inches per hour and was originally 0.5 inches tall, lasting approximately 4 minutes
  3. The candle loses 0.5 hours per inch burned and starts 8 hours long, lasting until length reaches zero
  4. The candle burns 0.5 inches per hour and was originally 8 inches tall, lasting exactly 16 hours (correct answer)
Explanation: When you encounter a linear equation like L=−0.5h+8L = -0.5h + 8, you need to interpret each component in context. The slope tells you the rate of change, and the y-intercept tells you the starting value. In this equation, the slope is -0.5, which means the length decreases by 0.5 inches for every hour of burning. The y-intercept is 8, representing the candle's original length when h=0h = 0. To find how long the candle lasts, set L=0L = 0: 0=−0.5h+80 = -0.5h + 8, so h=16h = 16 hours. This confirms answer choice D is correct. Let's examine why the other choices are wrong. Choice A misinterprets the slope as a rate (8 inches per 0.5 hours) and incorrectly calculates the original length as 16 inches instead of recognizing 8 as the y-intercept. Choice B confuses the variables entirely, treating the burning rate as 8 inches per hour and misinterpreting 0.5 as the original height. Choice C fundamentally misunderstands what the variables represent, treating hours as something that can be "lost per inch" and mixing up the units completely. The key trap here is misreading the slope and y-intercept. Remember that in y=mx+by = mx + b, the coefficient mm is always the rate of change, and bb is always the starting value. Pay careful attention to which variable is which, and always check your interpretation by substituting h=0h = 0 to verify the initial condition makes sense.

Question 4

A delivery truck's fuel efficiency follows the equation G = -0.08m + 15, where G is gallons remaining and m is miles driven. What does this relationship indicate about the truck's fuel consumption and operational constraints?

  1. Consumes 15 gallons per mile with 0.08-gallon tank, allowing extremely limited range for delivery operations
  2. Gains 0.08 gallons per mile with 15-gallon capacity, allowing unlimited range with efficient fuel recovery
  3. Uses 0.08 gallons per mile with 15 gallons initially, allowing approximately 187.5 miles maximum range (correct answer)
  4. Loses 0.08 miles per gallon used with 15-mile range, allowing minimal delivery distance per fuel unit
Explanation: When you encounter linear equations describing real-world relationships, focus on interpreting the slope and y-intercept in context. The equation G=−0.08m+15G = -0.08m + 15 is in slope-intercept form, where the slope represents the rate of change and the y-intercept represents the starting value. The coefficient -0.08 tells you the truck consumes 0.08 gallons per mile driven (the negative sign indicates fuel is being used up). The y-intercept of 15 means the truck starts with 15 gallons of fuel. To find maximum range, set G=0G = 0 and solve: 0=−0.08m+150 = -0.08m + 15, which gives m=187.5m = 187.5 miles. This confirms answer C is correct. Answer A misinterprets the slope and y-intercept, incorrectly stating 15 gallons per mile consumption and a 0.08-gallon tank capacity. Answer B impossibly suggests the truck gains fuel while driving, misreading the negative slope as positive and claiming unlimited range. Answer D confuses units entirely, treating the slope as "miles per gallon lost" rather than "gallons per mile," and misinterpreting the 15-gallon capacity as a 15-mile range. When analyzing linear equations in context, always identify what each variable represents, interpret the slope as a rate of change (positive means increasing, negative means decreasing), and remember the y-intercept gives you the starting condition. For optimization problems like finding maximum range, set the dependent variable to its limiting value and solve.

Question 5

A medication dosage model is D=150−8tD = 150 - 8t, where DD is the drug concentration in mg/L and tt is hours after administration. The medication becomes ineffective below 70 mg/L. What does the negative slope represent, and what timing constraint exists?

  1. Drug concentration increases by 8 mg/L per hour; effective for up to 10 hours
  2. Drug concentration decreases by 8 mg/L per hour; effective only at the 10-hour mark
  3. Drug concentration decreases by 8 mg/L per hour; effective for at most 10 hours (correct answer)
  4. Drug concentration decreases by 150 mg/L per hour; effective for at most 8 hours
Explanation: The slope -8 means concentration decreases by 8 mg/L per hour. Setting 70 = 150 - 8t gives t = 10 hours when it becomes ineffective, so it's effective for at most 10 hours (t ≤ 10). Choice A misinterprets the negative slope as positive. Choice B suggests effectiveness only at one specific moment rather than throughout a time period. Choice D misidentifies the slope as -150.

Question 6

A savings account balance follows B = 50w + 200, where B is the balance in dollars and w is weeks. What do the slope and y-intercept reveal about the account holder's financial behavior and initial situation?

  1. Deposits $200 monthly while maintaining $50 minimum, suggesting irregular saving with balance requirements
  2. Withdraws $50 weekly from a $200 maximum balance, suggesting spending habits exceeding income levels
  3. Earns $200 weekly with $50 in fees, suggesting high income offset by significant banking costs
  4. Saves $50 weekly with an initial $200 deposit, suggesting consistent saving habits and startup funds (correct answer)
Explanation: When you encounter a linear equation like B=50w+200B = 50w + 200, you're looking at a relationship where one variable changes at a constant rate relative to another. The key is understanding what the slope and y-intercept represent in the real-world context. In this savings account equation, the slope is 50, which tells you the rate of change per week. Since it's positive, the balance increases by $50 each week, meaning the account holder deposits $50 weekly. The y-intercept is 200, which represents the balance when $w=0w = 0 $ (at the starting point), so the account began with $200. Answer D correctly identifies both components: the person saves $50 weekly (the slope) and started with $200 (the y-intercept), indicating disciplined saving habits and initial startup funds. Answer A misinterprets the time unit as monthly instead of weekly and confuses the y-intercept as a minimum balance requirement rather than an initial amount. Answer B incorrectly treats the positive slope as withdrawals, which would actually decrease the balance and require a negative slope. Answer C confuses the y-intercept as weekly earnings and the slope as fees, completely reversing the mathematical relationship. Remember that in linear equations representing financial situations, always identify what the independent variable represents (here, weeks), then interpret the slope as the rate of change per unit of that variable, and the y-intercept as the starting value. The sign of the slope tells you whether values are increasing (positive) or decreasing (negative).

Question 7

A water reservoir's depth follows D = 0.25r - 12, where D is depth in feet and r is inches of rainfall. What do the slope and y-intercept indicate about the reservoir's current state and response to rainfall?

  1. Rainfall occurs at 0.25 inches per foot of depth; the reservoir is 12 inches deep initially
  2. Each inch of rain lowers depth by 0.25 feet; currently 12 feet above normal, requiring 48 inches to empty
  3. Each inch of rain raises depth by 0.25 feet; currently 12 feet below normal, requiring 48 inches to reach zero (correct answer)
  4. The reservoir gains 12 feet per 0.25 inches of rain; depth starts at negative values requiring immediate filling
Explanation: When you encounter a linear equation in the form D=mr+bD = mr + b, you're looking at how one variable changes with respect to another. Here, you need to interpret what the slope and y-intercept mean in the context of a water reservoir. The equation D=0.25r−12D = 0.25r - 12 tells us that for every inch of rainfall (r), the depth (D) increases by 0.25 feet. The positive slope of 0.25 means rainfall adds water to the reservoir. The y-intercept of -12 occurs when rainfall is zero (r=0r = 0), giving us D=−12D = -12 feet. This negative value indicates the reservoir starts 12 feet below some reference point (likely normal capacity). To find when the reservoir reaches zero depth, set D=0D = 0: 0=0.25r−120 = 0.25r - 12, so r=48r = 48 inches. This confirms answer C. Answer A misinterprets the slope direction and confuses the units—rainfall doesn't occur "per foot of depth." Answer B incorrectly states that rain lowers the depth, which contradicts the positive slope. While it correctly identifies needing 48 inches, it wrongly claims the reservoir is above normal. Answer D completely misreads both the slope (claiming 12 feet per 0.25 inches rather than 0.25 feet per inch) and misinterprets what "negative depth" means. Remember: in linear equations, a positive slope means the y-variable increases as the x-variable increases. Always check whether your interpretation makes logical sense in the real-world context—rain should add water to a reservoir, not remove it.

Question 8

A coffee shop's daily profit follows P = 12h - 180, where P is profit in dollars and h is hours of operation per day.

Based on the slope and y-intercept, what does this model reveal about the shop's operational efficiency and break-even requirements?

  1. Loses $12 per hour, starts with $180 profit, breaking even after exactly 15 hours of losses
  2. Earns $12 profit per hour, needs 15 hours daily to break even due to $180 in fixed costs (correct answer)
  3. Charges $180 per hour with $12 in daily expenses, requiring minimal operation for positive profits
  4. Generates $180 hourly revenue with $12 daily overhead, achieving immediate profitability upon opening
Explanation: When analyzing a linear profit equation like P = 12h - 180, you need to interpret the slope and y-intercept in the business context. The slope (12) represents the rate of change in profit per additional hour of operation, while the y-intercept (-180) shows the profit when hours equal zero. The slope of 12 means the shop earns $12 in profit for each hour it operates. The y-intercept of -180 indicates that with zero hours of operation, the shop has a loss of $180 - these are the fixed costs (rent, insurance, etc.) that occur regardless of whether the shop is open. To find the break-even point, set P = 0: $0=12h−1800 = 12h - 180 ,whichgives, which gives h=15h = 15 $ hours. Choice A incorrectly interprets the positive slope as a loss rather than a gain, and misunderstands what "breaking even" means. Choice C confuses the variables entirely, treating the y-intercept as an hourly charge and the slope as daily expenses. Choice D makes similar errors by interpreting -180 as positive hourly revenue and 12 as overhead costs, plus wrongly claiming immediate profitability. Choice B correctly identifies that the shop earns $12 profit per hour, faces $180 in fixed costs (explaining the negative y-intercept), and needs exactly 15 hours of daily operation to break even. Remember: in linear business models, always identify what the slope represents (rate of profit/loss per unit) and what the y-intercept means (fixed costs or initial conditions). The break-even point occurs where the function equals zero.

Question 9

A linear function modeling temperature change has the equation T = -3h + 72, where T is temperature in °F and h is hours after noon. Based on the slope and y-intercept, what can be concluded about the temperature conditions?

  1. Temperature drops 3°F hourly; at noon it was 72°F, indicating afternoon cooling from midday peak (correct answer)
  2. Temperature rises 3°F hourly; the minimum possible temperature is 72°F, indicating warming conditions
  3. Temperature varies by 72°F total; the rate is 3°F per hour, indicating moderate climate fluctuation
  4. Temperature stabilizes at 72°F; changes occur every 3 hours, indicating periodic weather pattern cycles
Explanation: The slope -3 means temperature decreases by 3°F each hour. The y-intercept 72 occurs when h = 0 (at noon), so the temperature at noon was 72°F. This suggests cooling from a midday high. Choice B ignores the negative slope. Choice C misinterprets what 72 represents (it's not total variation). Choice D misunderstands both the linear nature and what the coefficients represent.

Question 10

A smartphone battery drains according to B = -8t + 100, where B is battery percentage and t is hours of continuous use.

Considering both the slope and y-intercept, what constraint does this model place on maximum usage time, and what assumption does it make about the initial state?

  1. Maximum usage is 92 hours assuming the phone begins with 8% battery already consumed
  2. Maximum usage is 8 hours assuming the phone starts with 100 minutes of charge remaining
  3. Maximum usage is 100 hours assuming the battery drains at 8% per hour consistently
  4. Maximum usage is 12.5 hours assuming the phone starts fully charged at 100% battery (correct answer)
Explanation: When analyzing linear battery drain models, you need to interpret both the slope and y-intercept to understand the real-world constraints and assumptions. The equation B=−8t+100B = -8t + 100 tells us two key things: the y-intercept (100) represents the initial battery percentage when t=0t = 0, and the slope (-8) means the battery loses 8% per hour. To find maximum usage time, set B=0B = 0 (battery dead): 0=−8t+1000 = -8t + 100, so 8t=1008t = 100, giving us t=12.5t = 12.5 hours. The y-intercept of 100 means the phone starts fully charged at 100% battery. Looking at the wrong answers: Choice A incorrectly calculates maximum usage as 92 hours and misinterprets the y-intercept as showing 8% already consumed (the y-intercept actually shows the starting amount, not what's consumed). Choice B confuses units by mixing hours with minutes and gets the wrong maximum usage time of 8 hours. Choice C mistakes the slope value (-8) for the maximum usage time and incorrectly states maximum usage as 100 hours, mixing up the slope and y-intercept values. Choice D correctly identifies that maximum usage is 12.5 hours (when battery reaches 0%) and properly interprets the y-intercept of 100 as starting with a fully charged battery. Strategy tip: In linear models, always identify what the y-intercept represents (initial condition) and use algebra to find meaningful endpoints like when the dependent variable equals zero.

Question 11

An online subscription service tracks user engagement with the equation H = -2.5w + 40, where H represents average weekly hours of usage and w represents weeks since signup.

What does the negative slope indicate about user behavior, and what practical constraint does the y-intercept suggest about new subscribers?

  1. Usage decreases by 2.5 hours weekly; new users start with 40 hours, suggesting initial high engagement (correct answer)
  2. Usage increases by 2.5 hours weekly; the service costs $40 initially, suggesting a pricing barrier
  3. The service loses 2.5 users weekly; 40 represents initial subscriber count, suggesting growth challenges
  4. Costs decrease by $2.50 weekly; users save $40 total, suggesting a promotional discount structure
Explanation: The slope -2.5 means usage decreases by 2.5 hours each week. The y-intercept 40 occurs when w = 0 (at signup), indicating new users start with 40 hours of weekly usage, which is quite high and suggests initial enthusiasm. Choice B misses the negative slope and confuses H with cost. Choice C misinterprets H as user count rather than hours. Choice D completely misinterprets both variables.

Question 12

A population of bacteria grows according to P = 150t + 500, where P is the population and t is time in hours.

Based on the slope and y-intercept, what does this model assume about the growth pattern and initial laboratory conditions?

  1. Linear growth of 150 bacteria per hour starting from an initial culture of 500 bacteria (correct answer)
  2. Exponential growth with 150% hourly increase starting from a base population of 500 organisms
  3. Declining population losing 150 bacteria hourly from a maximum capacity of 500 total bacteria
  4. Cyclical growth varying by 500 bacteria every 150 hours according to natural population rhythms
Explanation: The equation is linear with slope 150 (bacteria added per hour) and y-intercept 500 (initial population when t = 0). This represents constant linear growth, not exponential. Choice B incorrectly describes exponential growth and percentage rates. Choice C makes the slope negative when it's positive. Choice D misinterprets the model as cyclical rather than linear and confuses the coefficients' meanings.

Question 13

A car rental company charges according to the equation C = 0.35m + 45, where C is the total cost in dollars and m is the miles driven. What does the slope tell us about the pricing structure, and what does this imply for customers?

  1. Each additional mile costs $0.35, implying customers pay more for longer trips regardless of base fees (correct answer)
  2. The total cost increases by $45 for every mile, implying distance is the primary cost factor
  3. Each mile reduces the cost by $0.35, implying longer trips become more economical per mile
  4. The cost per mile decreases as distance increases, implying bulk discounts for longer rentals
Explanation: The slope is 0.35, meaning for each additional mile driven, the cost increases by $0.35. This is a constant rate per mile added to the base fee. Choice B incorrectly identifies the y-intercept (45) as the slope. Choice C incorrectly suggests the slope is negative. Choice D incorrectly suggests a decreasing rate, but the slope is constant, not decreasing.

Question 14

An elevator's height follows h=20+3th = 20 + 3t for the first part of its journey, where hh is height in feet and tt is time in seconds. After reaching the 50-foot level, it travels at constant speed. How should the slope and domain restriction be interpreted?

  1. Elevator rises 3 feet per second; equation valid for 0≤t≤500 ≤ t ≤ 50 seconds
  2. Elevator rises 20 feet per second; equation valid for 0≤t≤100 ≤ t ≤ 10 seconds
  3. Elevator rises 3 feet per second; equation valid for 0≤t≤100 ≤ t ≤ 10 seconds (correct answer)
  4. Elevator rises 3 feet per second; equation valid for all positive values of tt
Explanation: When you encounter linear equations describing motion, you need to interpret both the slope (rate of change) and determine when the equation applies based on given conditions. The equation h=20+3th = 20 + 3t is in slope-intercept form, where the slope is 3. This means the elevator rises 3 feet per second. The y-intercept of 20 indicates the elevator starts at 20 feet when t=0t = 0. To find when the equation stops applying, you need to determine when the elevator reaches 50 feet. Setting h=50h = 50: 50=20+3t50 = 20 + 3t, so 30=3t30 = 3t, giving t=10t = 10 seconds. Since the elevator travels at constant speed after reaching 50 feet, this linear equation only describes the first 10 seconds of motion. Looking at the wrong answers: Choice A correctly identifies the 3 feet per second rate but incorrectly states the domain as 0≤t≤500 ≤ t ≤ 50 seconds. This confuses the height value (50 feet) with the time value (10 seconds). Choice B misinterprets the slope, thinking 20 is the rate of change rather than the starting height. While it correctly finds t=10t = 10, the fundamental misunderstanding of slope makes it wrong. Choice D correctly identifies the rate but ignores the constraint that the equation only applies until the elevator reaches 50 feet. Remember: in motion problems with piecewise behavior, always check when conditions change by solving for the boundary values. Don't confuse the units—here, 50 feet is a height, not a time.

Question 15

The temperature of a cooling object follows the equation T=180−12tT = 180 - 12t, where TT is temperature in degrees Fahrenheit and tt is time in minutes. If the object reaches room temperature (72°F) and stops cooling further, what constraint does this place on the domain of this linear model?

  1. The model is valid for t≤9t ≤ 9 minutes only (correct answer)
  2. The model is valid for t≤15t ≤ 15 minutes only
  3. The model is valid for t≥9t ≥ 9 minutes only
  4. The model is valid for all real values of tt
Explanation: Setting T = 72: 72 = 180 - 12t, so 12t = 108, giving t = 9 minutes. Since the object stops cooling at room temperature, the linear model only applies until it reaches 72°F at t = 9 minutes. Choice B uses the wrong calculation (180/12 = 15). Choice C reverses the inequality direction. Choice D ignores the physical constraint.

Question 16

A water tank is being drained for maintenance. The equation V=1200−15tV = 1200 - 15t represents the volume of water VV (in gallons) remaining in the tank after tt minutes of draining. If the tank must retain at least 300 gallons to prevent damage to the pump, what does the slope of this equation represent in this context?

  1. The tank drains at a rate of 15 gallons per minute (correct answer)
  2. The tank fills at a rate of 15 gallons per minute
  3. The tank started with 15 gallons of water initially
  4. The tank will be empty after exactly 15 minutes
Explanation: The slope is -15, which represents the rate of change of volume with respect to time. Since the slope is negative, the volume is decreasing at 15 gallons per minute, meaning the tank drains at 15 gallons per minute. Choice B incorrectly interprets the negative slope as positive. Choice C confuses slope with y-intercept. Choice D misinterprets what the slope represents.

Question 17

A company's profit model is given by P=45n−2800P = 45n - 2800, where PP represents profit in dollars and nn represents the number of units sold. Based on this model, what is the significance of the y-intercept in the business context?

  1. The company makes a profit of $2800 when no units are sold
  2. The company has fixed costs of $2800 that must be covered before making profit (correct answer)
  3. The company must sell exactly 2800 units to break even
  4. The company makes $2800 profit per unit sold after covering expenses
Explanation: When n = 0, P = -2800, meaning the company has a loss of $2800 when no units are sold. This represents fixed costs that must be overcome. Choice A incorrectly states this as a profit rather than a loss. Choice C confuses the y-intercept with the x-intercept (break-even point). Choice D confuses the y-intercept with the slope.

Question 18

A car rental company charges according to the formula C=35+0.25mC = 35 + 0.25m, where CC is the total cost in dollars and mm is miles driven. A customer's budget allows for a maximum cost of $85. What does the coefficient 0.25 represent, and how does it affect the customer's driving limit?

  1. It's the base rental fee; the customer can drive at most 200 miles within budget
  2. It's the cost per mile; the customer can drive at most 200 miles within budget (correct answer)
  3. It's the cost per mile; the customer can drive at most 140 miles within budget
  4. It's the hourly rate; the customer can drive at most 240 miles within budget
Explanation: The coefficient 0.25 is the slope, representing the cost per mile ($0.25). Setting 85 = 35 + 0.25m gives 50 = 0.25m, so m = 200 miles maximum. Choice A correctly finds 200 miles but misidentifies what 0.25 represents. Choice C correctly identifies 0.25 as cost per mile but calculates incorrectly (using 85/0.25 instead of (85-35)/0.25). Choice D misinterprets the units and miscalculates.

Question 19

A spring's length changes according to L=8+0.5FL = 8 + 0.5F, where LL is length in inches and FF is applied force in pounds. The spring breaks if it exceeds 14 inches. What do the slope and y-intercept tell us about this spring's physical properties?

  1. Natural length is 8 inches; spring extends 0.5 inches per pound; maximum safe force is 14 pounds
  2. Natural length is 8 inches; spring extends 2 inches per pound; maximum safe force is 12 pounds
  3. Natural length is 0.5 inches; spring extends 8 inches per pound; maximum safe force is 12 pounds
  4. Natural length is 8 inches; spring extends 0.5 inches per pound; maximum safe force is 12 pounds (correct answer)
Explanation: When you encounter a linear equation describing a physical relationship, interpret it using the slope-intercept form y=mx+by = mx + b, where the y-intercept represents the starting condition and the slope shows the rate of change. In L=8+0.5FL = 8 + 0.5F, the y-intercept is 8, which represents the spring's length when no force is applied (F=0F = 0). This is the natural length: 8 inches. The slope is 0.5, meaning the spring extends 0.5 inches for each additional pound of force. To find the maximum safe force, set the length equal to the breaking point: 14=8+0.5F14 = 8 + 0.5F. Solving: 6=0.5F6 = 0.5F, so F=12F = 12 pounds. Choice A incorrectly states the maximum safe force is 14 pounds, confusing the breaking length (14 inches) with the force measurement. Choice B claims the spring extends 2 inches per pound, which would be the reciprocal of the actual slope—this reverses the relationship between force and extension. Choice C completely misinterprets the equation, treating the slope (0.5) as the natural length and the y-intercept (8) as the extension rate per pound. Choice D correctly identifies all three properties: natural length of 8 inches, extension rate of 0.5 inches per pound, and maximum safe force of 12 pounds. Study tip: In linear equations describing physical systems, always identify what happens at zero input (y-intercept) and the rate of change (slope). Then use algebra to find critical values like maximum limits.

Question 20

A battery's voltage decreases linearly according to V=12−0.4tV = 12 - 0.4t, where VV is voltage and tt is time in hours. The device stops working when voltage drops to 8 volts. What does the slope indicate about battery performance, and what operational constraint does this create?

  1. Battery loses 0.4 volts per hour; device operates for only the final hour before failure
  2. Battery gains 0.4 volts per hour; device operates indefinitely
  3. Battery loses 12 volts per hour; device operates for at most 0.4 hours
  4. Battery loses 0.4 volts per hour; device operates for at most 10 hours (correct answer)
Explanation: When you encounter a linear equation like V=12−0.4tV = 12 - 0.4t, you're looking at a relationship where one variable changes at a constant rate with respect to another. The slope tells you exactly how fast that change occurs. In this equation, the slope is −0.4-0.4, which means the voltage decreases by 0.4 volts for every hour that passes. The negative sign indicates a decrease, so the battery loses 0.4 volts per hour. To find when the device stops working, set the voltage equal to 8: 8=12−0.4t8 = 12 - 0.4t. Solving this gives 0.4t=40.4t = 4, so t=10t = 10 hours. This means the device can operate for at most 10 hours before the voltage drops too low. Choice A correctly identifies the voltage loss rate but incorrectly claims the device only works for the final hour. The device actually works for the entire 10-hour period until failure. Choice B gets the direction wrong—the negative slope means voltage decreases, not increases, and the device definitely doesn't operate indefinitely. Choice C confuses the y-intercept (12) with the slope and dramatically underestimates the operating time. The correct answer is D: the battery loses 0.4 volts per hour, and the device operates for at most 10 hours. Remember that in linear equations, the coefficient of the variable is always the slope, and its sign tells you the direction of change. When solving real-world constraint problems, set up an equation where the limiting condition equals your linear expression.