1.6 Approximation and error questions for IB Maths AI

Calculate the GDP per person of Kestavia. Give your answer to the nearest dollar.

Write your answer to part (a) in the form $a\times {10}^k$, where $1\le a<10$ and $k\in \mathbb{Z}$, correct to three significant figures.

In the same year, the GDP of the neighbouring country Luvenor was $2.6\times {10}^{11}$ USD.

Calculate how many times greater the GDP of Kestavia was than the GDP of Luvenor.

Calculate the volume of the tank, in m³.

Given that 1 m³ $=$ 1000 litres, write down the volume of the tank in litres. Give your answer in the form $a\times {10}^k$, where $1\le a<10$ and $k\in \mathbb{Z}$.

The town uses $1.35\times {10}^7$ litres of water each day. No water enters the tank.

Find the number of complete days that a full tank can supply the town.

The radius and the depth of the tank were each measured correct to the nearest metre.

Find the upper bound of the volume of the tank.

The lower bound of the volume of the tank is 94 685.6 m³, correct to one decimal place.

Find the largest possible percentage error in using your answer to part (a) as the volume of the tank.

During one storm, 12 mm of rain fell on the area that drains into the tank. This area is 3.6 km².

(i) Find the volume of rain, in m³, that fell on this area. Give your answer in the form $a\times {10}^k$, where $1\le a<10$ and $k\in \mathbb{Z}$.

(ii) Find this volume as a percentage of the volume of the tank found in part (a).

By letting $\omega =\frac{\mathrm{d}\theta }{\mathrm{d}t}$, write the differential equation as a system of first-order equations in the form $\left(\begin{array}{c}\dot{\theta }\\ \dot{\omega }\end{array}\right)=\mathit{M}\left(\begin{array}{c}\theta \\ \omega \end{array}\right)$, where $\mathit{M}$ is a $2\times 2$ matrix in terms of $c$.

The closer is first set with $c=5$.

Show that the eigenvalues of $\mathit{M}$ are $-1$ and $-4$.

Find an eigenvector for each eigenvalue.

Hence show that $\theta =1.6{\mathrm{e}}^{-t}-0.4{\mathrm{e}}^{-4t}$.

The door is considered shut once $\theta <0.05$.

Find the time taken for the door to be shut.

Show that, with this setting, the door never swings past its closed position.

Sketch the phase portrait for this system, with $\theta$ on the horizontal axis and $\omega$ on the vertical axis. Your sketch should show the eigenvector lines and the trajectory that starts at $(1.2,0)$.

After some years the closer wears and the damping setting falls to $c=2$.

Find the eigenvalues of $\mathit{M}$ for this setting.

Hence describe the motion of the door with this setting.

Use Euler's method with a step length of 0.1 seconds to estimate the value of $\theta$ when $t=0.5$ for the worn closer.

A computer solution of the differential equation gives $\theta =0.7916$ when $t=0.5$.

Calculate the percentage error in your answer to part (h).

Find the set of values of $c$ for which $\mathit{M}$ has two distinct real eigenvalues.

Explain why the manufacturer would not choose a very large value of $c$.

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 ($t=3$).

Calculate the volume of one red blood cell, in m³, according to Ingrid's model. Give your answer in the form $a\times {10}^k$, where $1\le a<10$ and $k\in \mathbb{Z}$.

An adult has 5.0 litres of blood, and 45% of the volume of the blood is made up of red blood cells. 1 litre $={10}^{-3}$ m³.

Estimate the number of red blood cells in the adult's body. Give your answer in the form $a\times {10}^k$, where $1\le a<10$ and $k\in \mathbb{Z}$.

A medical textbook states that the adult has $2.5\times {10}^{13}$ red blood cells.

Taking the textbook value as exact, calculate the percentage error in Ingrid's estimate.

Each red blood cell contains $2.7\times {10}^8$ haemoglobin molecules.

Using your answer to part (b), find the total number of haemoglobin molecules in the adult's red blood cells.

Oxygen is absorbed through the surface of the red blood cells.

Calculate the total surface area of one red blood cell according to Ingrid's model.

Real red blood cells are thinner in the middle than at the edge.

State whether Ingrid's model is likely to overestimate or underestimate the volume of one red blood cell.

Use the trapezoidal rule with all the values in the table to estimate the distance between the two stops.

The tram company's engineer, Mireille, models the velocity by

Find the distance between the stops according to this model.

Find the percentage difference between your answer to part (a) and your answer to part (b)(i), taking the value in part (b)(i) as the true value.

Use differentiation to find the maximum velocity of the tram according to Mireille's model.

Find the maximum acceleration of the tram according to the model, and the time at which it occurs.

For passenger comfort, the magnitude of the acceleration of a tram must never exceed 1.3 m s⁻².

Show that Mireille's model satisfies this condition.

Find the time at which the tram has travelled half of the distance between the stops.

On a new route, two stops are 900 m apart. Mireille uses a model of the same form,

where $T$ seconds is the journey time and $c$ is a positive constant.

Show that $c={\displaystyle \frac{10800}{T^4}}$.

Show that the greatest magnitude of the acceleration during the journey is ${\displaystyle \frac{10800}{T^2}}$.

Hence find the shortest journey time on the new route that satisfies the comfort condition in part (e).

Find the maximum velocity of the tram on this journey.

State one reason why Mireille's model may not be suitable for the new route.