2.6 The modelling process questions for IB Maths AI

For each situation, write down the most appropriate type of model.

Give a reason for your choice of model for situation A.

Give a reason for your choice of model for situation B.

Find the cost of each plan for a month in which Zara uses 8 GB of data.

Write down a piecewise function $A(x)$ for the cost of Plan A, for $x\ge 0$.

Find the amount of data for which Plan A and Plan B cost the same, where $x<5$.

Determine the range of values of $x$ for which Plan A is the cheapest of the three plans.

Zara's monthly data use varies between 4 GB and 12 GB. State, with a reason, which plan you would recommend.

Find the equation of the regression line of $P$ on $v$.

Write down the coefficient of determination, $R^2$, for this linear model.

Calculate the residuals of the linear model at $v=4$, $v=8$ and $v=12$, and hence explain why a linear model is not appropriate, despite the high value of $R^2$.

Tomasz next considers a power model $P=a{v}^b$.

Show that this model gives a linear relationship between $\ln P$ and $\ln v$.

Use your graphic display calculator to find the values of $a$ and $b$.

For a model $P=f(v)$, the sum of square residuals is $S{S}_{res}={\sum }_{i=1}^9(P_i-f(v_i){)}^2$.

Find the value of $S{S}_{res}$ for the power model.

Physical theory suggests that the power output is proportional to the cube of the wind speed, so Tomasz now considers the model $P=k{v}^3$. He chooses $k$ to minimize

Show that $S(k)$ is minimized when $k={\displaystyle \frac{\sum v_i^3{P}_i}{\sum v_i^6}}$.

Hence find the value of $k$.

For the model $P=k{v}^3$, $S{S}_{res}=516$, correct to three significant figures.

Tomasz notices that this one-parameter model has a smaller value of $S{S}_{res}$ than the power model in part (b), which has two parameters. Explain why this is possible, and state which of the two models fits the data better.

On one day, the wind speed at the site $t$ hours after midnight is modelled by $v=7+3\sin \left(\frac{\pi t}{12}\right)$, for $0\le t\le 24$.

The energy, in kWh, produced by the turbine in one day is ${\int }_0^{24}P\mathrm{d}t$. Use the model $P=k{v}^3$ to find the energy produced that day.

The turbine's maximum power output is 1500 kW. Comment on using the model $P=k{v}^3$ to predict the power output when the wind speed is 15 m s⁻¹.

Give a reason why a model of this form is appropriate for these data.

Using the maximum and minimum values in the table, find the value of $a$ and the value of $d$.

Write down the value of $b$.

Calculate the percentage error when Francesca's model is used to estimate the mean temperature in April ($t=3$).

(i) Write down the value of $V(0)$.

(ii) Interpret this value in context.

Find the number of views after 7 days.

Find the number of days until the video reaches 50 million views.

Show that, according to this model, the number of views doubles approximately every two days.

Explain why this model is not suitable for predicting the number of views after several months.

An analyst proposes a second model,

(i) Write down the equation of the horizontal asymptote of the graph of $W$.

(ii) Interpret your answer to part (f)(i) in context.

(i) Find the value of $t$, $t>0$, at which the two models predict the same number of views.

(ii) Hence suggest how the two models could be combined to describe the number of views.

Show that $h={\displaystyle \frac{500}{\pi r^2}}-{\displaystyle \frac{2r}{3}}$.

Show that the total cost, $C$ USD, of building the silo is given by

Explain why the model is only valid for $0<r<6.20$.

Find ${\displaystyle \frac{\mathrm{d}C}{\mathrm{d}r}}$.

Find the value of $r$ and the value of $h$ that minimize the cost, and find this minimum cost to the nearest dollar.

Use the second derivative to justify that this value of $r$ gives a minimum cost.

A local planning regulation states that the total height of the silo, including the roof, must not exceed 9 m.

Show that the radius must satisfy ${\displaystyle \frac{500}{\pi r^2}}+{\displaystyle \frac{r}{3}}\le 9$.

Find the least value of $r$ that satisfies the regulation.

Explain why the cheapest silo that satisfies the regulation has the radius found in part (g)(ii), and find its cost to the nearest dollar.

Find the percentage increase in the minimum cost caused by the regulation.

Suggest one further cost that the cooperative should include in the model before deciding on the radius of the silo.