Linear Functions: Interpolation

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.

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 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?

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.

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).

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.

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.

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 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.

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.

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.

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.

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.

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.

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.