IB Mathematics: Applications and Interpretation Quiz: Linear Functions And Models

20 questions · exam conditions

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Linear Functions And ModelsQuestion 1 of 20

A company's annual profit has been increasing linearly. In 2016, the profit was $2.5 million. In 2021, the profit was $4.0 million. Based on this trend, what was the profit in 2018?

$3.0 million

$3.4 million

$3.25 million

$3.1 million

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IB Mathematics: Applications and Interpretation Quiz

IB Mathematics: Applications and Interpretation Quiz: Linear Functions And Models

Practice Linear Functions And Models in IB Mathematics: Applications and Interpretation 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 Linear Functions And Models, giving you a quick way to practice the rules, question types, and explanations that matter most for IB Mathematics: Applications and Interpretation.

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.

All questions

Question 1

A company's annual profit has been increasing linearly. In 2016, the profit was $2.5 million. In 2021, the profit was $4.0 million. Based on this trend, what was the profit in 2018?

$3.0 million
$3.4 million
$3.25 million
$3.1 million (correct answer)

Explanation: When you encounter a problem about linear growth or change over time, you're working with a straight-line relationship where the rate of change remains constant. This means you can find the slope and use it to predict values at any point. First, calculate the rate of change. From 2016 to 2021 (5 years), profit increased from $2.5 million to $4.0 million, so the change was $1.5 million over 5 years. This gives a rate of $

\frac{1.5}{5} = 0.3

$ million dollars per year. To find the 2018 profit, note that 2018 is 2 years after 2016. Starting from the 2016 profit of $2.5 million and adding 2 years of growth:

2.5 + (2 \times 0.3) = 2.5 + 0.6 = 3.1

million dollars. Looking at the wrong answers: Choice A (

3.0 million) represents the exact midpoint between 2016 and 2021 profits, but 2018 isn't the midpoint year—that would be 2018.5. Choice B (

3.4 million) likely comes from miscalculating the rate as $0.45 million per year instead of

0.3 million. Choice C (

3.25 million) represents the profit exactly halfway between the two given values, ignoring that 2018 is closer to 2016 than to 2021. The correct answer is D) $3.1 million. Strategy tip: For linear growth problems, always calculate the rate per unit time first, then multiply by the time elapsed from your reference point. Don't assume the target year is at the midpoint unless the math confirms it.

Question 2

A company's annual profit has been increasing linearly. In 2016, the profit was $2.5 million. In 2021, the profit was $4.0 million. Based on this trend, what was the profit in 2018?

$3.0 million
$3.4 million
$3.25 million
$3.1 million (correct answer)

\frac{1.5}{5} = 0.3

$ million dollars per year. To find the 2018 profit, note that 2018 is 2 years after 2016. Starting from the 2016 profit of $2.5 million and adding 2 years of growth:

2.5 + (2 \times 0.3) = 2.5 + 0.6 = 3.1

million dollars. Looking at the wrong answers: Choice A (

3.0 million) represents the exact midpoint between 2016 and 2021 profits, but 2018 isn't the midpoint year—that would be 2018.5. Choice B (

3.4 million) likely comes from miscalculating the rate as $0.45 million per year instead of

0.3 million. Choice C (

Question 3

The temperature in a freezer is -18°C. When the power is turned off, the temperature increases at a constant rate. After 40 minutes, the temperature is -10°C. If the relationship is linear, how many minutes will it take for the freezer to reach 0°C?

90 minutes (correct answer)
80 minutes
50 minutes
100 minutes

Explanation: This is a linear relationship problem where temperature changes at a constant rate over time. When you see phrases like "constant rate" and "linear relationship," you're dealing with a straight-line equation where you need to find the slope and use it to predict future values. First, find the rate of temperature increase. The temperature rises from -18°C to -10°C in 40 minutes, so the change is

-10 - (-18) = 8°C

in 40 minutes. This gives a rate of

\frac{8°C}{40 \text{ min}} = 0.2°C \text{ per minute}

. To reach 0°C from the starting temperature of -18°C, you need a total temperature increase of

0 - (-18) = 18°C

. At a rate of 0.2°C per minute, this takes

\frac{18°C}{0.2°C/\text{min}} = 90 \text{ minutes}

. This confirms answer A. Looking at the wrong answers: B (80 minutes) would only allow for a 16°C increase, leaving you at -2°C. C (50 minutes) represents a common error where students might add just 10 minutes to the given 40 minutes without proper calculation. D (100 minutes) would result in a 20°C increase, overshooting to 2°C. For linear relationship problems, always identify the rate of change first, then determine the total change needed to reach your target value. Remember that when working with negative temperatures, moving toward zero means the values are increasing (becoming less negative).

Question 4

The speed of a car decreases linearly after the brakes are applied. Two seconds after applying the brakes, the car's speed is 24 m/s. Five seconds after applying the brakes, its speed is 15 m/s. What was the car's speed at the instant the brakes were applied?

30 m/s (correct answer)
28 m/s
27 m/s
33 m/s

Explanation: When you encounter a problem about linear change over time, you're working with a constant rate of change, which means you can model the situation with a linear equation. Since the car's speed decreases linearly, you can find the rate of change first. Between 2 seconds and 5 seconds (a 3-second interval), the speed drops from 24 m/s to 15 m/s, giving you a decrease of 9 m/s over 3 seconds. This means the rate of change is

\frac{-9}{3} = -3 \text{ m/s per second}

. Now you can work backwards from either known point. Using the 2-second mark: if the car was going 24 m/s after 2 seconds of braking, and speed decreases by 3 m/s each second, then 2 seconds earlier (at

t = 0

), the speed was

24 + (2 \times 3) = 24 + 6 = 30 \text{ m/s}

. Looking at the wrong answers: B) 28 m/s would give you 22 m/s at 2 seconds, not 24 m/s. C) 27 m/s would result in 21 m/s at 2 seconds. D) 33 m/s would mean 27 m/s at 2 seconds. Each of these fails to match the given data point. Therefore, A) 30 m/s is correct. Study tip: For linear motion problems, always identify your rate of change first, then use it consistently to work forward or backward in time. Double-check your answer by verifying it produces both given data points.

Question 5

A company produces custom t-shirts. The cost to produce 50 shirts is $450. The cost to produce 120 shirts is $910. The relationship between the number of shirts produced, n, and the total cost, C, is linear.

A company produces custom t-shirts. The cost to produce 50 shirts is $455. The cost to produce 120 shirts is $910. The relationship between the number of shirts produced, n, and the total cost, C, is linear.

$6.50
$9.10
$120.00
$130.00 (correct answer)

Explanation: The fixed cost is the y-intercept of the linear cost function C(n) = mn + b. We have two points: (50, 455) and (120, 910). First, calculate the variable cost per shirt (the gradient m): m = (910 - 455) / (120 - 50) = 455 / 70 = $6.50 per shirt. Now use one point to find the fixed cost b. Using (50, 455): 455 = 6.50(50) + b 455 = 325 + b b = 130 Therefore, the company's fixed cost is $130.

Question 6

The battery percentage, B, of a smartphone after t hours of continuous video playback is modelled by the linear function B(t) = 100 - 8t.

Which of the following statements correctly interprets the y-intercept of this function?

The smartphone battery is fully charged after 100 hours of use.
The smartphone starts with a 100% battery charge before use. (correct answer)
The battery life decreases by 100% for every hour of use.
The smartphone can be used for a maximum of 100 hours.

Explanation: The y-intercept of a function f(x) is the value of f(x) when x=0. In this model, B(t) = 100 - 8t, the y-intercept occurs at t=0. B(0) = 100 - 8(0) = 100. Since t represents hours of use, t=0 represents the starting time before any use. B(t) is the battery percentage. Therefore, the y-intercept value of 100 means the smartphone starts with a 100% battery charge.

Question 7

A car was purchased for a certain price. Its value depreciates linearly over time. After 2 years, its value is $24,000. After 5 years, its value is $19,500. What was the initial purchase price of the car?

$21,000
$27,000 (correct answer)
$28,500
$33,000

Explanation: Let V(t) be the value of the car after t years. The model is V(t) = mt + c, where c is the initial price. We have two points: (2, 24000) and (5, 19500). The rate of depreciation (gradient m) is m = (19500 - 24000) / (5 - 2) = -4500 / 3 = -$1500 per year. The model is V(t) = -1500t + c. To find the initial price c (the value at t=0), we can use one of the points. Using (2, 24000): 24000 = -1500(2) + c 24000 = -3000 + c c = 27000. The initial price was $27,000.

Question 8

The cost, C, in euros, to rent a conference room for h hours is given by the function C(h) = 70h + 200. A company was charged a total of 655 euros for renting the room. For how many hours did the company rent the room?

5.5 hours
12.2 hours
9.4 hours
6.5 hours (correct answer)

Explanation: This question tests your ability to solve linear equations in a real-world context. When you see a cost function like

C(h) = 70h + 200

, recognize that you're dealing with a linear relationship where 70 is the hourly rate and 200 is a fixed cost (like a setup fee). To find how many hours the company rented the room, you need to solve for h when the total cost C equals 655 euros. Set up the equation:

655 = 70h + 200

. Subtract 200 from both sides:

455 = 70h

. Then divide by 70:

h = \frac{455}{70} = 6.5

hours. Let's examine why the other answers are wrong. Choice A (5.5 hours) would give a cost of

70(5.5) + 200 = 585

euros, which is too low. Choice B (12.2 hours) would cost

70(12.2) + 200 = 1054

euros, far exceeding the actual charge. Choice C (9.4 hours) would result in

70(9.4) + 200 = 858

euros, also too high. These incorrect answers likely represent common algebraic mistakes: perhaps forgetting to subtract the fixed cost first, making arithmetic errors, or confusing the setup of the equation. Study tip: For linear cost functions, always identify the variable cost (coefficient of h) and fixed cost (constant term) first. Then work systematically: substitute the known total, isolate the variable term, and solve. Always verify your answer by plugging it back into the original equation.

Question 9

The value of an investment, V, in USD, is modelled by the function V(t) = 5000 + 250t, where t is the number of years after the initial investment. What is the percentage increase in the value of the investment from the end of year 3 to the end of year 8?

20.0%
21.7% (correct answer)
25.0%
30.0%

Explanation: First, calculate the value of the investment at t=3 and t=8. V(3) = 5000 + 250(3) = 5000 + 750 = 5750. V(8) = 5000 + 250(8) = 5000 + 2000 = 7000. Next, find the absolute increase in value: Increase = V(8) - V(3) = 7000 - 5750 = 1250. Finally, calculate the percentage increase relative to the value at the start of the period (t=3): Percentage increase = (Increase / V(3)) * 100% = (1250 / 5750) * 100% ≈ 21.739%. Rounded to one decimal place, this is 21.7%.

Question 10

A company manufactures a product with a total cost function C(x) = 15x + 3000, where x is the number of units produced. The company sells each unit for $35. What is the minimum number of units that must be sold to make a profit?

86
150
151 (correct answer)
200

Explanation: To make a profit, total revenue must be greater than total cost. The revenue function is R(x) = 35x. The break-even point occurs when R(x) = C(x). 35x = 15x + 3000 20x = 3000 x = 3000 / 20 = 150. At 150 units, the company breaks even (profit is zero). To make a profit, the company must sell more than 150 units. The minimum integer number of units to make a profit is 151.

Question 11

A mobile mechanic charges a fixed call-out fee plus an hourly rate. A 2-hour job costs a total of $195, and a 5-hour job costs a total of $435. What is the fixed call-out fee?

$35 (correct answer)
$45
$80
$87

Explanation: Let C be the total cost and t be the time in hours. The model is linear: C(t) = mt + b, where m is the hourly rate and b is the fixed call-out fee. We have two points: (2, 195) and (5, 435). First, find the hourly rate (gradient m): m = (435 - 195) / (5 - 2) = 240 / 3 = $80 per hour. Now substitute one point into C(t) = 80t + b to find the fixed fee b. Using (2, 195): 195 = 80(2) + b 195 = 160 + b b = 35 Therefore, the fixed call-out fee is $35.

Question 12

A salesperson earns a fixed monthly salary plus a commission that is a constant percentage of their total sales for the month. In April, they had total sales of $30,000 and earned a total of $3,500. In May, they had total sales of $45,000 and earned a total of $4,250. What is their fixed monthly salary?

$1,500
$1,750
$2,000 (correct answer)
$2,250

Explanation: Let E be the total earnings and S be the total sales. The linear model is E(S) = mS + b, where m is the commission rate and b is the fixed salary. We have two points: (30000, 3500) and (45000, 4250). The gradient m (commission rate) is m = (4250 - 3500) / (45000 - 30000) = 750 / 15000 = 0.05. The commission rate is 5%. Now, substitute one of the points into E = 0.05S + b to find the fixed salary b. Using (30000, 3500): 3500 = 0.05(30000) + b 3500 = 1500 + b b = 3500 - 1500 = 2000. The fixed monthly salary is $2,000.

Question 13

A water tank is being filled at a constant rate. The volume of water in the tank, V litres, after t minutes is given by the function V(t) = 12t + 50. How long does it take for the volume of water to increase from 290 litres to 830 litres?

45 minutes (correct answer)
50 minutes
54 minutes
65 minutes

Explanation: We need to find the time t1 when V(t1) = 290 and the time t2 when V(t2) = 830. The duration is t2 - t1. For V = 290: 290 = 12t1 + 50 => 240 = 12t1 => t1 = 20 minutes. For V = 830: 830 = 12t2 + 50 => 780 = 12t2 => t2 = 65 minutes. The time taken for the increase is t2 - t1 = 65 - 20 = 45 minutes. Alternatively, the total increase in volume is 830 - 290 = 540 litres. The rate of filling is the gradient, which is 12 litres/minute. The time required is Total Volume Change / Rate = 540 / 12 = 45 minutes.

Question 14

The battery percentage, B, of a smartphone after t hours of continuous video playback is modelled by the linear function B(t) = 100 - 8t.

Which of the following statements correctly interprets the y-intercept of this function?

The smartphone battery is fully charged after 100 hours of use.
The smartphone starts with a 100% battery charge before use. (correct answer)
The battery life decreases by 100% for every hour of use.
The smartphone can be used for a maximum of 100 hours.

Question 15

A diver's depth below the water surface, D in metres, is modelled by the function D(t) = 2t - 6 for t ≥ 3, where t is the time in seconds after leaving a diving board. What is the correct interpretation of the value t = 3 in this model?

The diver jumps from a height of 3 metres.
The diver reaches their maximum depth after 3 seconds.
The model is valid starting from the moment the diver enters the water. (correct answer)
The diver starts descending into the water at a rate of 3 m/s.

Explanation: The model is defined for t ≥ 3. Let's find the depth at t=3: D(3) = 2(3) - 6 = 0. A depth of 0 metres represents the water surface. Since the model begins at t=3 where the depth is 0, t=3 represents the time the diver enters the water. The model describes the diver's depth from that moment onwards.

Question 16

A taxi fare is calculated using the formula F(d) = 2.50 + 1.80d, where F is the total fare in dollars and d is the distance travelled in kilometres. The company policy states the maximum distance for a single trip is 25 km. What is the range of possible fares for a trip?

$2.50 to $45.00
$1.80 to $47.50
$1.80 to $45.00
$2.50 to $47.50 (correct answer)

Explanation: When you encounter a function with a restricted domain, you need to find the range by evaluating the function at the endpoints of that domain. Here, you have a linear fare function

F(d) = 2.50 + 1.80d

where the distance

d

is constrained between 0 and 25 kilometers. Since this is a linear function with a positive slope (1.80), it's strictly increasing. This means the minimum fare occurs at the smallest distance and the maximum fare occurs at the largest distance. At the minimum distance (

d = 0

F(0) = 2.50 + 1.80(0) = \$2.50

At the maximum distance (

d = 25

F(25) = 2.50 + 1.80(25) = 2.50 + 45.00 = \$47.50

Therefore, the range is $2.50 to $47.50, making D correct. Looking at the wrong answers: A gives the correct minimum but caps the maximum at

45.00, which represents only the variable cost portion (

1.80 \times 25

) without adding the base fare. B incorrectly uses

1.80 as the minimum fare, confusing the per-kilometer rate with the actual minimum fare, while also using the wrong maximum from option A. C combines both errors from A and B: wrong minimum ($1.80) and wrong maximum ($45.00). Study tip: For linear functions on restricted domains, always evaluate at both endpoints to find the range. The base value in the function (here, $2.50) represents your minimum output when the variable equals zero.

Question 17

$1,500
$1,750
$2,000 (correct answer)
$2,250

Question 18

45 minutes (correct answer)
50 minutes
54 minutes
65 minutes

Question 19

The population of a small town, P, has been growing linearly since the year 2010. In 2015, the population was 8,400. In 2021, the population was 9,720.

Based on this linear model, what is the predicted population of the town in 2025?

10,280
10,400
10,600 (correct answer)
10,840

Explanation: Let t be the number of years since 2010. So, 2015 corresponds to t=5, and 2021 corresponds to t=11. We want to find the population in 2025, which is t=15. The two data points are (5, 8400) and (11, 9720). The rate of change (gradient m) is m = (9720 - 8400) / (11 - 5) = 1320 / 6 = 220 people per year. We can find the population in 2025 by extrapolating from the population in 2021. The year 2025 is 4 years after 2021. Predicted population = Population in 2021 + (Rate of change) * (Number of years) P(15) = 9720 + 220 * (15 - 11) = 9720 + 220 * 4 = 9720 + 880 = 10600.

Question 20

Two mobile phone plans are available. Plan A costs $40 per month plus $0.10 per minute of calls. Plan B costs $25 per month plus $0.15 per minute of calls.

For how many minutes of calls per month would the total cost of the two plans be exactly the same?

150 minutes
260 minutes
300 minutes (correct answer)
433 minutes

Explanation: Let m be the number of minutes. The cost functions for the two plans are: Plan A: C_A(m) = 40 + 0.10m Plan B: C_B(m) = 25 + 0.15m To find when the costs are the same, we set C_A(m) = C_B(m): 40 + 0.10m = 25 + 0.15m Subtract 25 from both sides: 15 + 0.10m = 0.15m Subtract 0.10m from both sides: 15 = 0.05m Divide by 0.05: m = 15 / 0.05 = 300. The cost is the same for 300 minutes of calls.