Exercises: Linear functions: interpolation

Look at the following graph, which represents the total cost of an order of custom T-shirts for an e-sports team as a function of the number of units purchased.

From the graph:

1. a) Identify the y-intercept ($n$) and explain its meaning in this context.

1. b) If the total cost for $10$ T-shirts is $50$ pesos, calculate the slope ($m$) of the line.

1. c) Write the analytical expression of the function $f(x)$.

An e-scooter rental company charges a fixed unlock fee and a constant rate for each minute of use. One user paid $90$ pesos for a $15$-minute trip, while another paid $150$ pesos for a $35$-minute trip.

Find the equation of the linear function relating the cost of the service ($y$) to the time of use in minutes ($x$).

The following table shows the temperature of an industrial oven as it heats up at a constant rate:

Use linear interpolation to estimate the temperature of the oven when $12$ minutes have elapsed since it was switched on.

A gardener observes that the growth of a species of bamboo is linear during its first few weeks. If at $10$ days it measured $45$ cm and at $20$ days it measured $75$ cm:

1. Determine the linear function that models the height of the bamboo as a function of time.

2. Perform a linear extrapolation to predict the height of the plant at $25$ days.

Find the linear function passing through the points $A(2, 7)$ and $B(5, 16)$. Once obtained, check whether the point $C(4, 13)$ belongs to this function using linear interpolation.

Two music platforms offer the following monthly plans:

• Plan A: $5$ pesos fixed fee plus $0.20$ pesos for each song downloaded.

• Plan B: No fixed fee, but $0.60$ pesos for each song downloaded.

Let $x$ be the number of songs downloaded in a month and $y$ the monthly cost in pesos.

1. a) Write the cost function for each plan.

1. b) Graph both functions on the same Cartesian axes (it is enough to take values of $x$ between $0$ and $20$ songs) and find graphically the "break-even point" (where both plans cost the same). State its coordinates.

It is known that the relationship between the braking distance of a car ($d$) and its speed ($v$) is approximately linear for certain low-speed ranges. If at $20$ km/h the braking distance is $4$ meters and at $40$ km/h it is $12$ meters:

1. What would be the estimated braking distance for a speed of $30$ km/h?

2. If a car has left a braking mark of $10$ meters, approximately how fast was it traveling? (Use interpolation).

Given the following table of values relating the number of hours worked by a freelance programmer and their total fees:

Calculate, using the formula of the linear interpolation line, how much they would charge for a job of $7$ hours. Also state the hourly rate charged (gradient).

Check that the parabola $y = x^{2} - 2x + 3$ passes through the points $(0, 3)$, $(1, 2)$ and $(2, 3)$, and use it to estimate the value of $y$ at $x = 1.5$.

We want to interpolate the points $(0, 1)$, $(1, 4)$ and $(2, 9)$ using a parabola $y = ax^{2} + bx + c$.

1. a) Set up the system of three equations in unknowns $a$, $b$ and $c$.

1. b) Check that $a = 1$, $b = 2$ and $c = 1$ is a solution to the system and write the interpolating parabola.

A streaming content platform has analyzed the growth of its subscribers in a young city. In 2021 it had $12\,400$ users, and in 2023 the figure rose to $15\,200$.

If we assume that the growth follows a linear model, and we take $x$ as the time in years elapsed since 2021 (that is, $x=0$ for the year 2021, $x=2$ for 2023, and so on):

1. a) Find the linear interpolation function that expresses the number of subscribers as a function of time $x$.

1. b) Use interpolation to estimate the number of subscribers in mid-2022 (that is, for $x=1.5$).

1. c) Carry out an extrapolation to predict how many users there will be in 2026 and reflect on the reliability of this last figure.

The figure shows the two known data points (note that the vertical axis starts at $12\,000$).

A biology laboratory studies the breakdown of an organic compound. It has been observed that at a temperature of $15^\circ\text{C}$, the compound takes $40$ hours to decompose, whereas at $25^\circ\text{C}$ it takes only $28$ hours.

1. Determine the linear interpolation line relating the decomposition time ($y$) to the temperature ($x$).

2. What would be the estimated decomposition time if the temperature were $22^\circ\text{C}$?

3. If the laboratory needs the compound to degrade in $20$ hours, at what temperature should the thermostat be set according to this model?

The organisers of an urban music festival have observed the relationship between ticket price and the number of season tickets sold. If the price is $45$ pesos, $3\,200$ season tickets are sold. If the price rises to $60$ pesos, sales drop to $2\,450$ season tickets.

1. Find the equation of demand assuming it is a linear function.

2. Estimate how many season tickets would be sold if an "intermediate" price of $52$ pesos were set.

3. Above what price would demand be zero according to this extrapolation model?

The percentage of plastic packaging recycling in an autonomous region has evolved almost linearly over the last decade. In the year 2012, $18\,\%$ was recycled, and in the year 2020, $34\,\%$ was reached.

1. Calculate the mean annual rate of change of the recycling percentage between those years.

2. Use linear interpolation to estimate the percentage of recycling in the year 2015.

3. If the trend continues in the same way, in which year is the European Union target of $50\,\%$ expected to be reached?

An electric car consumes battery energy proportionally to the distance traveled. Leaving the garage with $100\,\%$ charge, after traveling $120$ km, the battery shows $64\,\%$ charge remaining.

1. Express the linear function indicating the percentage of battery remaining as a function of kilometers traveled.

2. If the driver decides to stop and recharge when $15\,\%$ battery remains, how many kilometers will they have traveled in total according to interpolation?

3. Using extrapolation, determine what battery percentage it would show after a trip of $300$ km. Does the result obtained make physical sense? Justify your answer.

In a sociological study on the use of digital devices, the following data were collected on the daily screen time (in minutes) spent by teenagers according to their age:

1. Determine the linear function that fits this data.

2. Estimate, by interpolation, the daily screen time for a $14$-year-old teenager.

3. If a $18$-year-old youth spends $250$ minutes a day, does this data point fit the linear model calculated above? Check it using extrapolation.

The total cost of producing custom t-shirts for a graduation includes a fixed set-up cost for the design and a variable cost per unit manufactured. We know that $50$ t-shirts cost $8\,200$ pesos and that $120$ t-shirts cost $16\,600$ pesos.

1. Calculate the cost of manufacturing $85$ t-shirts using the linear interpolation technique.

2. Identify from the resulting line the fixed cost (y-intercept) and the cost per additional t-shirt (gradient).

3. If the class's maximum budget is $12\,000$ pesos, what is the maximum number of t-shirts they can order?

The snow depth at a ski resort is being measured during a day of continuous snowfall. At $8:00$ h the depth was $45$ cm and at $13:00$ h it had risen to $62$ cm.

1. Find the equation of the line relating snow depth ($y$) to the hours elapsed since $8:00$ h ($x$).

2. What snow depth was estimated at $11:30$ h?

3. If the snowfall began at $5:00$ h in the morning with the same intensity, how much snow was on the ground at that time? Use linear extrapolation.

The figure shows a parabola passing through the points $(1, 3)$, $(3, 7)$ and $(5, 5)$.

1. a) Set up the system of equations that $a$, $b$ and $c$ must satisfy so that $y = ax^{2} + bx + c$ passes through the three points, and solve it.

1. b) Use the parabola obtained to estimate the value of $y$ at $x = 4$ (the one marked with "?" in the figure).

1. c) Check with the graph that the estimate is consistent.

A concert promoter has tested three ticket prices with different results: at $\$10$ it sold $400$ tickets; at $\$20$, $700$; and at $\$30$, $600$. The figure shows the data and the interpolating parabola.

1. a) Find the parabola $y = ax^{2} + bx + c$ that interpolates the three points (price, tickets sold).

1. b) Estimate how many tickets it would sell at $\$25$.

1. c) According to the model, which price maximises the number of tickets sold? Calculate the vertex of the parabola and interpret the result.

The table shows the first values of a sequence:

• $n$: $1$, $2$, $3$, $4$, $5$

• $a_n$: $4$, $9$, $18$, $31$, $48$

1. a) Calculate the first differences (each term minus the previous one) and the second differences (differences of the differences). What do you observe in the second differences?

1. b) Justify why if the second differences are constant, the sequence follows a quadratic model $a_n = an^{2} + bn + c$, and find that model.

1. c) Calculate $a_6$ using the model and check it by extending the differences.

A sustainable courier company uses electric bicycles for its deliveries. It has been observed that the monthly maintenance cost per bicycle depends on the distance traveled in kilometers. If a bicycle travels $400\text{ km}$, the cost is $\$2\,400$, whereas if it travels $1000\text{ km}$, the cost rises to $\$4\,200$.

1. a) Using linear interpolation, determine the analytic expression of the function relating the monthly cost $C(x)$ to the distance traveled $x$.

1. b) If the maximum maintenance budget per bicycle is $\$3\,600$, what is the estimated maximum distance in kilometers that it can travel according to this model?

1. c) The company plans to expand and estimates that some bicycles will travel $1500\text{ km}$ per month. Calculate the cost using linear extrapolation and reason whether this estimate is reliable in the long term based on physical wear and tear of the components.

The market value of a specific model of high-end smartphone depreciates linearly during the first few months after its launch. 2 months after its release, its price was $\$19\,000$. At 8 months, its price dropped to $\$14\,200$.

1. a) Calculate, using linear interpolation, the price of the device 5 months after its launch.

1. b) Determine the original launch price ($x = 0$) using extrapolation.

1. c) If a user decides to sell their smartphone when its value is exactly half of the original price, in which month should they put it up for sale according to this model?

A sociological study analyses the relationship between the number of hours per week spent on digital leisure and the score on an emotional well-being test (from 0 to 100). It is known that for 10 hours of leisure the score is 85 points, and for 35 hours the score drops to 60 points.

1. a) Obtain the linear interpolation line that models this situation.

1. b) A group of students has an average score of 72 points. Estimate how many hours of digital leisure they consume per week.

1. c) If a person consumes no digital leisure ($x = 0$), what score would be assigned to them? What if they consume 80 hours? Discuss the validity of the results at the extremes of the function.

An NGO working on the reforestation of a burned area has recorded the growth of a type of pine tree. 3 years after transplanting, the average height was $1.8\text{ m}$. At 7 years, the height reached $3.4\text{ m}$.

1. a) Assuming linear growth, estimate the height of the pine trees at 5 and a half years.

1. b) The forestry team considers the pine trees to be "adults" when they exceed $5\text{ meters}$. How many years must pass from transplanting for them to reach that height according to linear extrapolation?

1. c) If the pine tree measured $40\text{ cm}$ at the time of transplanting ($t = 0$), check whether the linear model from the previous sections fits this initial data point. If not, explain what could be occurring biologically.

The gas bill for a building community has a variable charge that depends on consumption in $kWh$. In January, with a consumption of $2500\text{ kWh}$, the total bill (including a standing charge) was $\$6\,200$. In February, consumption dropped to $1800\text{ kWh}$ and the bill was $\$4\,660$.

1. a) Calculate the price per $kWh$ and the value of the standing charge on the bill using linear interpolation.

1. b) Estimate the bill for a consumption of $3000\text{ kWh}$.

1. c) Due to a new local authority tax, the standing charge will increase by $15\%$ and the price per $kWh$ by $5\%$. Write the new cost function and compare it graphically with the previous one.

Two quantities $X$ and $Y$ are linearly related. It is known that for $x_1 = 12$ we get $y_1 = 40$, and that for $x_2 = 20$ we get $y_2 = 30$. The figure shows these two data points and the region in which the relationship is valid.

1. a) Determine the value of $y$ for $x = 15$ using linear interpolation.

1. b) Find the value of $x$ for which $y$ vanishes ($y = 0$). ¿Pertenece ese valor a la zona de validez?

1. c) If this linear relationship is known to be valid only on the interval $x \in [10, 30]$, calculate the maximum and minimum values that the quantity $Y$ can take on that interval.

The supply and demand laws for a product have been modelled linearly using the following data collected from a market research study:

• At a price of $\$300$, $200$ units are supplied and $500$ are demanded.

• At a price of $\$500$, $400$ units are supplied and $300$ are demanded.

1. a) Calculate the supply function $O(p)$ and demand function $D(p)$ using linear interpolation.

1. b) Find the equilibrium price (where supply equals demand) and the equilibrium quantity.

1. c) If the government imposes a maximum price of $\$360$, will there be excess supply or a shortage of demand? Calculate the exact difference in units.

A marathon runner maintains a steady pace during a stretch of his training. At kilometer 5 of his route, the stopwatch read 22 minutes and 30 seconds. At kilometer 12, the elapsed time was 54 minutes.

1. a) Express the time $T$ (in minutes) as a function of the distance $d$ (in kilometers) assuming that the pace is constant.

1. b) Calculate, using interpolation, at which kilometer the runner was when the stopwatch read exactly 45 minutes.

1. c) If the runner started the marathon at kilometer 0, how long did it take him to complete the first 500 meters? Use extrapolation and express the result in minutes and seconds.

The figure shows the points $(0, 2)$, $(1, 5)$ and $(2, 10)$, the line interpolating the first two, and the parabola interpolating all three.

1. a) Find the linear interpolation line passing through $(0, 2)$ and $(1, 5)$.

1. b) Find the parabola $y = ax^{2} + bx + c$ passing through all three points.

1. c) Calculate the prediction of each model for $x = 3$ and their difference. If the data were the amount of an electricity bill that grows faster and faster across consumption brackets, which model seems more reliable to you? Reason your answer.

Prove that through three points with distinct x-coordinates $(x_1, y_1)$, $(x_2, y_2)$ and $(x_3, y_3)$ passes a unique polynomial function of degree less than or equal to $2$.

1. a) Set up the system of three equations with unknowns $a$, $b$ and $c$ resulting from requiring $y = ax^{2} + bx + c$ to pass through the three points.

1. b) Justify that the system has a unique solution by calculating its determinant (or discussing the system) and using the fact that the x-coordinates are distinct.

1. c) Explain why "degree less than or equal to $2$" includes the possibility of the solution being a straight line: what condition would the three points then satisfy?

On a summer day, temperatures in a city were measured: at $8$ h, $12$ °C; at $14$ h, $24$ °C; at $20$ h, $18$ °C.

1. a) Taking $t = 0$ at $8$ h, find the parabola $T(t) = at^{2} + bt + c$ interpolating the three data points.

1. b) According to the model, at what time was the maximum temperature reached, and what was its value?

1. c) What temperature does the model predict for $2$ h the following morning ($t = 18$)? Is it reasonable? Comment on the limitations of interpolation when used to extrapolate outside the data range.