1.1 Scientific notation questions for IB Maths AI

Write down the distance from the Earth to the Sun in the form $a\times {10}^k$, where $1\le a<10$ and $k\in \mathbb{Z}$.

Write down the speed of light correct to three significant figures, in the form $a\times {10}^k$, where $1\le a<10$ and $k\in \mathbb{Z}$.

Calculate the time, in minutes, that light takes to travel from the Sun to the Earth.

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).

Dr Okafor wants to plot the data. Suggest why she should use a logarithmic scale for the mass.

Find the values of ${\log }_{10}M$ and ${\log }_{10}H$ for the horse.

Let $X={\log }_{10}M$ and $Y={\log }_{10}H$.

Find Pearson's product-moment correlation coefficient, $r$, for $X$ and $Y$. Hence comment on whether a model of the form $H=k{M}^b$ is suitable.

Find the equation of the regression line of $Y$ on $X$.

Use your answer to part (d) to show that $H=k{M}^b$, and find the value of $k$ and the value of $b$.

A giraffe has a mass of 800 kg and a measured resting heart rate of 48 beats per minute.

Use your model to estimate the resting heart rate of the giraffe, and find the percentage error in this estimate.

Find the percentage decrease in resting heart rate that this model predicts when body mass is doubled.

Find the mass of a mammal with a predicted resting heart rate of 150 beats per minute.

Dr Okafor models the mean lifespan, $L$ years, of a mammal by $L=8M^{0.2}$. She investigates the total number of heartbeats, $N$, in a mammal's lifetime, assuming that the heart beats at its resting rate throughout. A year has 525 600 minutes.

Show that $N\approx 1.02\times {10}^9\times M^{-0.058}$.

Find the value of $N$ for a mouse and for an elephant. Give your answers in the form $a\times {10}^k$, where $1\le a<10$ and $k\in \mathbb{Z}$.

A popular science article claims that "every mammal gets about one billion heartbeats in its lifetime". Comment on this claim, with reference to your answers to parts (i)(i) and (i)(ii).

A blue whale has a mass of about 150 000 kg. State one reason why Dr Okafor's model may not give a reliable estimate of its heart rate.

Write down the diameter of the virus particle in the form $a\times {10}^k$, where $1\le a<10$ and $k\in \mathbb{Z}$.

Mateo imagines placing these virus particles in a straight line, touching each other, across a fingernail of width 2.5 cm.

Calculate the number of virus particles needed. Give your answer in the form $a\times {10}^k$, where $1\le a<10$ and $k\in \mathbb{Z}$.

Write down the distance between the centre of the Earth and the centre of the Moon in metres, in the form $a\times {10}^k$, where $1\le a<10$ and $k\in \mathbb{Z}$.

Calculate the gravitational force between the Earth and the Moon. Give your answer in the form $a\times {10}^k$, where $1\le a<10$ and $k\in \mathbb{Z}$.

A weather satellite of mass 1850 kg orbits the Earth. The gravitational force between the Earth and the satellite is $4.6\times {10}^3$ N.

Find the distance, $r$, between the centre of the Earth and the satellite.

The radius of the Earth is $6.37\times {10}^6$ m.

Find the height of the satellite above the surface of the Earth, in km.

The speed of the satellite, $v$ m s⁻¹, is given by $v=\sqrt{{\displaystyle \frac{GM}{r}}}$, where $M$ is the mass of the Earth.

Calculate the speed of the satellite.

The satellite travels in a circle around the centre of the Earth.

Find the time, in hours, that the satellite takes to complete one orbit.

Write the populations of country A and of country B in the form $a\times {10}^k$, where $1\le a<10$ and $k\in \mathbb{Z}$.

List the four countries in order of population, from smallest to largest.

Find the total population of the four countries. Give your answer in the form $a\times {10}^k$, where $1\le a<10$ and $k\in \mathbb{Z}$, correct to three significant figures.