1.1 Scientific notation questions for IB Maths AA

Find the volume of one cube, giving your answer in scientific notation.

The jeweller melts down a gold bar of volume $1.28\times {10}^{-4}$ m${}^3$ to make the cubes. No gold is lost.

Find the number of cubes made from the bar.

Find the radius of the tank.

Find the capacity of the tank in litres, giving your answer in the form $a\times {10}^k$, with $1\le a<10$ and $k\in \mathbb{Z}$.

Write down the magnitude of an earthquake for which $A={10}^5{A}_0$.

Earthquake X has magnitude $6.5$ and earthquake Y has magnitude $4.5$. The amplitudes recorded for X and Y are $A_X$ and $A_Y$ respectively.

Find ${\displaystyle \frac{A_X}{A_Y}}$.

The energy, $E$ joules, released by an earthquake of magnitude $M$ is given by ${\log }_{10}E=1.5M+4.8$.

Show that earthquake X releases $1000$ times as much energy as earthquake Y.

Show that $E={10}^{4.8}{\left({\displaystyle \frac{A}{A_0}}\right)}^{\frac{3}{2}}$.

Earthquake Z releases $8$ times as much energy as earthquake W.

Use part (d) to find the ratio of the amplitude recorded for Z to the amplitude recorded for W.

Given that ${\log }_{10}2=0.301$, correct to three significant figures, find the difference between the magnitudes of earthquakes Z and W.

Find the sound level of a sound with intensity $3.2\times {10}^{-5}$ W m${}^{-2}$.

At a concert hall, sounds have intensities in the interval ${10}^{-12}\le I\le 10$. Find the range of $L$ for these intensities.

Find an expression for $L^{-1}(x)$.

Hence find the intensity of a sound with a sound level of $85$ dB, writing your answer in scientific notation.

Show that $L^{-1}(x+10)=10\times L^{-1}(x)$.

A sound is produced by a single loudspeaker. A second, identical loudspeaker is then switched on, which doubles the intensity of the sound. Find the increase in the sound level.

Find the total mass, in grams, of the bacteria in the sample. Write your answer in the form $a\times {10}^k$, with $1\le a<10$ and $k\in \mathbb{Z}$.

Write down the total mass of the bacteria in kilograms, in the form $a\times {10}^k$, with $1\le a<10$ and $k\in \mathbb{Z}$.

A second sample of bacteria of the same type has a total mass of $1.5$ grams.

Find the number of bacteria in the second sample. Write your answer in the form $a\times {10}^k$, with $1\le a<10$ and $k\in \mathbb{Z}$.

Apply Euler's method, with a step length of $0.2$, to estimate the value of $y$ at $x=1.6$.

Use the substitution $y=vx$ to show that $x{\displaystyle \frac{\mathrm{d}v}{\mathrm{d}x}}=v^2$.

Hence, by solving the differential equation, show that $y={\displaystyle \frac{x}{2-\ln x}}$.

Find the percentage error in the estimate found in part (a).

Write down the equation of the vertical asymptote of the solution curve.

Find the exact coordinates of the point on the solution curve at which the gradient is 2.