Scientific notation: notes and practice questions

The Richter scale measures the magnitude of an earthquake using the formula:

$M = \log_{10}\left( \frac{I}{I_{0}} \right)$

Where $M$ is the magnitude, $I$ is the intensity of the earthquake, and $I_{0}$ is a reference intensity.

$a$ Find the magnitude of an earthquake with an intensity $7.5 \times 10^{4}$ times greater than the reference intensity $I_{0}$.

(b.i) Write an expression for I in terms of $M$ and $I_{0}$

(b.ii) Find the intensity of an earthquake with magnitude 5.6, given the reference intensity $I_{0} = 10^{- 12}\ Wm^{- 2}.$ Give your answer in the form $b \times 10^{a}$, where $1 \leq b < 10$ and $a$ is an integer

A pharmaceutical company is estimating its vaccine production for the next quarter. The initial projection for a new vaccine batch is 859 000 000 doses.

The exact number of doses produced in this batch turns out to be 876 543 210.

(a) Find the percentage error in the company's projection.

The company plans to produce seven identical batches of this vaccine.

(b) (i) Find the total number of doses needed for these seven batches, to three significant figures.

(ii) Write down your answer to part (b)(i) in the form $a \times 10^k$, where $1 \le a < 10$, $k \in \mathbb{Z}$.

A tech company launches a new social media app. The number of active users, $U$, can be modelled by the function

$U(t) = 800 \times k^t, t\ge 0$,

where $t$ is the number of hours since the app was launched, and $k$ is a positive constant.

Write down the value of $U(0)$.

Interpret what this value means in this context.

4 hours after the app was launched, the number of active users was 4050.

Find the value of $k$.

Find the number of active users 2 hours and 15 minutes after the app was launched.

A competitor launches a similar app, whose user base, $U_2$, can be modelled by the function

$U_2(t) = 2500 \times 1.2^t, t\ge 0$,

where $t$ is the number of hours since both apps were launched.

Find the value of $t$ when the number of users for both apps is equal.

It takes $H$ hours and $m$ minutes for the number of users of the first app to reach 15000.

Find the value of $H$ and $m$, giving $m$ as an integer.

Each user of the first app requires $1.5 \times 10^{-2}$ MB of server storage. The total available server capacity is $3.0 \times 10^5$ MB.

Determine how long it would take for the app's user base to exceed the server capacity.

A spherical ice sculpture is melting in a gallery. Initially, the sculpture has a radius of 30 cm. This information is illustrated in the following diagram.

The gallery curator predicts that, as the ice sculpture melts, its radius will decrease at a constant rate of 0.5 cm per hour.

According to this model, find

(i) the radius of the ice sculpture, 10 hours after it begins melting.

(ii) the volume of the ice sculpture, 10 hours after it begins melting. Give your answer to one decimal place.

Let the function $V(t)$ represent the volume of the ice sculpture, $\text{cm}^3$, $t$ hours after it begins melting. $V(t)$ is given by

$V(t) = 113000 - 5000t + 150t^2 - 1.5t^3$, for $0 \le t \le 20$.

Find $V'(t)$.

Find the rate of change of the volume of the ice sculpture at $t = 10$ hours.

State one reason why the radius of the ice sculpture may not always decrease at a constant rate.

A large water wheel is used for irrigation. The lowest point a bucket on the wheel reaches is $0.5$ m above the water surface, and its highest point is $12.5$ m above the water surface.

(a) (i) Show that the radius of the water wheel is $6$ m.

(ii) Calculate the circumference of the water wheel.

(b) When the wheel rotates $15^\circ$, find the distance that a bucket travels along the circumference.

The height in metres, above the water surface, of a particular bucket is modelled by the function:

$h(t) = a \sin (bt) + d$, for $a, b > 0$,

where $t$ is the time, measured in minutes.

The wheel takes $12$ minutes to complete $1$ revolution.

(c) (i) Find the value of $b$.

(ii) Find the value of $d$.

(iii) Hence, write down the equation of the sinusoidal model.

(d) Use the model to find the values of $t$ when the height of this bucket is $9$ m above the water surface for $0 \le t \le 12$.

The water wheel operates for $2500$ days, and each day it rotates nonstop for $10$ hours.

(e) Calculate the total number of rotations that the water wheel has made. Give your answer in the form $a \times 10^k$ where $1 \le a < 10$ and $k$ is an integer.

The following table shows the student enrollment figures for various faculties at a major university.

(a) Calculate the total number of students enrolled at the university.

(b) Determine the percentage of students enrolled in the Faculty of Engineering, giving your answer to one decimal place.

(a) A catering company offers custom-designed cakes for events.

For an order of 10 cakes or fewer, the price per cake is $P$ dollars (USD). For larger orders, the price of each cake after the first 10 ordered is reduced by 8 USD.

$C(n)$ represents the total cost of purchasing $n$ cakes from the company.

Write down an expression in terms of $P$ for the total cost of ordering 10 cakes.

(b) Write down an equation for $C(n)$, in terms of $P$ and $n$, for the total cost of ordering $n$ cakes when $n > 10$.

(c.i) A client orders 80 cakes for a corporate event. The mean price per cake for this order is $32.15$ USD.

Calculate the exact total cost of the order.

(c.ii) Hence, determine the value of $P$, correct to 2 decimal places.

(a) The magnitude of an earthquake, $M$, on the Richter scale, is determined by the amplitude of its seismic waves, $A$, measured in micrometers ($\mu$m). The relationship between magnitude and amplitude can be expressed using the logarithmic function

$M = \log_{10}\left(\frac{A}{A_0}\right)$, $A>0$

where $A_0$ is the reference amplitude (a very small, baseline amplitude).

The reference amplitude $A_0$ is $10^{-6}$ $\mu$m.

An earthquake is recorded with a seismic wave amplitude of $10^{1.5}$ $\mu$m.

Calculate the magnitude of this earthquake on the Richter scale.

(b) Another earthquake has a magnitude of $6.2$ on the Richter scale.

Determine the amplitude of its seismic waves. Give your answer in the form $a \times 10^k$ where $1 \le a < 10$, $k \in \mathbb{Z}$.

A new environmental monitoring system measures the Severity Index $S$ of a certain pollutant using the formula $S = 10 \log_{10} \left(\frac{P}{P_0}\right)$, where $P$ is the concentration of the pollutant in micrograms per cubic meter ($\mu g/m^3$) and $P_0 = 10^{-3} \mu g/m^3$ is a reference concentration.

(a) Calculate the Severity Index $S$ when the pollutant concentration $P$ is $100 \mu g/m^3$.

(b) The Severity Index for a different pollutant is recorded as $95$ units. Determine the concentration $P$ of this pollutant. Give your answer in the form $a \times 10^k$, where $1 \le a < 10$ and $k \in \mathbb{Z}$.

GreenGrow company designs modular garden beds. Each garden bed is constructed from individual square panels. The company offers different configurations, where adding more 'sections' increases both the length and width of the garden bed. When $N$ sections are added, the garden bed has a total length of $L_N$ centimetres and a total width of $W_N$ centimetres.

The following table shows the values of $L_N$ and $W_N$ for the first three values of $N$.

(a.i) Find the value of $a$.

(a.ii) Find the value of $b$.

(b.i) Write down an expression in terms of $N$ for $L_N$.

(b.ii) Write down an expression in terms of $N$ for $W_N$.

(c.i) A customer wants a garden bed with a total length of 440 cm. Each section of the garden bed requires 6 individual panels. Show that the customer needs 120 panels.

(c.ii) Find the value of $W_N$ for this garden bed.

(d) Each individual panel is a square with side length 25 cm. Find the total area of the panels in the garden bed. Give your answer in the form $a \times 10^k$ where $1 \le a < 10$ and $k$ is an integer.

(e) The panels cost $18.00 per square metre and are sold in packs of 10 panels.

Find the cost of a single pack of 10 panels.

(f) To allow for damaged panels, the customer wants at least 15% more panels than needed.

Find the minimum number of packs of panels the customer will need to order.

(g) There is a fixed delivery cost of $40.

Find the total cost for the customer's order.

A scientist estimates that a large bacterial culture contains approximately $543210000000000000000000000$ individual bacteria.

Write down this number in the form $a \times 10^k$ where $1 \le a < 10$ and $k \in \mathbb{Z}$.

A large spherical weather balloon is designed with a radius of $12.5$ metres.

Calculate the estimated volume of the balloon, in $\text{m}^3$. Give your answer to three significant figures.

The actual volume of the weather balloon is later measured to be $8200\text{ m}^3$.

Find the percentage error in the estimated volume calculated in part (b). Give your answer to three significant figures.