Scientific notation: notes and practice questions

(a) A newly proposed particle accelerator, the "Cosmic Collider", is designed with a perfectly circular beam path. The engineers have specified its diameter as 15 000 metres.

Calculate the circumference of the Cosmic Collider's beam path in kilometres. Give your full calculator display.

(b) Give your answer to part (a) correct to three significant figures.

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

A high-performance computing cluster transfers data at a constant rate of 250 megabytes per second (MB/s). A large scientific dataset, with a total size of 1200 gigabytes (GB), needs to be transferred.

(a) Calculate the time, in minutes, it takes for the computing cluster to transfer the entire dataset. Assume 1 GB = 1000 MB.

A research facility stores experimental data. Each experiment generates 3.5 terabytes (TB) of data. The facility plans to conduct 8500 such experiments over the next few years.

(b) Find the total storage capacity, in megabytes (MB), required for all these experiments. Give your answer in the form $a \times 10^{k}$ with $1 \le a < 10$ and $k \in \mathbb{Z}$ . Assume 1 TB = 1000 GB and 1 GB = 1000 MB.

A pharmaceutical company is formulating a new drug. They are testing two active ingredients, $x_1$ and $x_2$, such that their total concentration is $24$ mg/mL. The drug's efficacy is modelled by the product of the concentrations of the two ingredients.

Find the efficacy, $E$, as a function of $x_1$ only.

Determine the concentration of $x_1$ that maximizes the drug's efficacy.

Hence, show that the maximum efficacy for two ingredients with a total concentration of $24$ mg/mL is $144$.

Let $M_n(S)$ represent the maximum efficacy for a drug with $n$ active ingredients and a total concentration of $S$ mg/mL. For $n = 2$, the maximum efficacy can be expressed as $M_2(S) = \left(\frac{S}{2}\right)^2$.

Verify that $M_2(S) = \left(\frac{S}{2}\right)^2$ is true for $S = 24$.

The relationship between the geometric mean and arithmetic mean states that for $n$ positive real numbers $x_1, x_2, ..., x_n$, their geometric mean $(x_1 \times x_2 \times ... \times x_n)^{\frac{1}{n}}$ is always less than or equal to their arithmetic mean $\frac{x_1 + x_2 + ... + x_n}{n}$.

Show that the geometric mean and arithmetic mean are equal when $x_1 = x_2 = ... = x_n$.

Use this result to prove that $M_n(S) = \left(\frac{S}{n}\right)^n$.

Using the formula for $M_n(S)$, determine the value of:

$M_3(24)$;

$M_4(24)$;

$M_5(24)$.

For a fixed total concentration of $S = 24$ mg/mL, the company wants to find the optimal number of active ingredients, $n$, to maximize the drug's efficacy. Let $P(S)$ denote this maximum efficacy.

Write down the value of $P(24)$ and the value of $n$ at which it occurs.

Determine the value of $P(30)$ and the value of $n$ at which it occurs.

Consider the continuous function $h$, defined by $\ln(h(x) ) = x\ln\left(\frac{S}{x}\right)$, where $x \in \mathbb{R}^+$. A sketch of the graph of $y = h(x)$ is shown in the following diagram. Point A is the maximum point on this graph.

Find, in terms of $S$, the $x$-coordinate of point A.

Verify that $h(x) = M_x(S)$, when $x \in \mathbb{Z}^+$.

The company has a total concentration of $S = 150$ mg/mL available. Use your answer to part (h) to find the largest possible efficacy. Give your answer in the form $a \times 10^k$, where $1 \le a < 10$ and $k \in \mathbb{Z}^+$.

A scientist is studying a spherical microbe. The radius of the microbe is measured to be $3 \times 10^{-5}$ cm.

(a) Write down the diameter of the microbe.

(b) The volume of the microbe can be expressed in the form $\pi(a \times 10^k) \text{ cm}^3$ where $1 \leq a < 10$ and $k \in \mathbb{Z}$.

Find the value of $a$ and the value of $k$.

Consider a geometric sequence, $u_n$, with common ratio $r$, where each term in the sequence is positive.

It is given that $u_3 = 3.2 \times 10^2$ and $u_5 = 5.12 \times 10^3$.

(a) Find the value of $r$.

(b) Find the value of $u_1$, giving your answer in the form $a \times 10^k$, where $1 \le a < 10$ and $k \in \mathbb{Z}$.

The following table shows the average annual temperature, $x$ in degrees Celsius, and the average annual sunshine hours, $y$, for seven different cities.

The city of Sunville has an average annual temperature of 18 °C.

In fact, Sunville has an average of 2650 annual sunshine hours.

The city of Frostburg has an average annual temperature of -2 °C.

(a) Find the range of the average annual temperatures for these seven cities.

(b) (i) For the data from these seven cities, calculate $r$, the Pearson’s product–moment correlation coefficient.

(ii) Describe the correlation between the average annual temperature and the average annual sunshine hours.

(c) Write down the equation of the regression line $y$ on $x$, in the form $y = mx + c$.

(d) Use your regression line to estimate the average annual sunshine hours for Sunville.

(e) Find the percentage error in your estimate in part (d).

(f) State whether it is valid to use the regression line to estimate the average annual sunshine hours for Frostburg. Give a reason for your answer.

A large, rare pearl is discovered, and its value is estimated to be 250 million US dollars ($USD$).

(a) Write down this value in the form $a \times 10^k$, where $1 \le a < 10$ and $k \in \mathbb{Z}$

The pearl is assumed to be a perfect sphere with a radius of 3 cm.

(b) Calculate an estimate for the volume of the pearl, giving your answer in terms of $\pi$.

The actual volume of the pearl is later determined to be $38\pi \text{ cm}^3$.

(c) Calculate the percentage error in the estimated volume.

A large conical sand dune has a height of $9 \times 10^2$ m and a base radius of $2 \times 10^3$ m.

(a) Find the area of the base of the sand dune, giving your answer in terms of $\pi$.

The volume of the sand dune can be expressed in the form $V = \pi(b \times 10^n)$ m$^3$, where $1 \le b < 10$ and $n \in \mathbb{Z}$.

(b) Find the value of $b$ and the value of $n$.

A space probe is sent to a dwarf planet. The total distance of the journey is $1.8 \times 10^9$ kilometres. The probe travels at an average speed of 50 000 kilometres per hour.

(a) Calculate the time, in years, it takes for the probe to complete its journey. Assume 1 year = 9000 hours.

The probe is designed to collect interstellar dust particles. The average mass of one dust particle is 0.4 nanograms (ng). During its journey, the probe collects 50 million particles.

(b) Find the total mass of the dust particles collected, in kilograms (kg). Give your answer in the form $a \times 10^{k}$ where $1 \le a < 10$ and $k \in \mathbb{Z}$. Assume 1 gram = $10^9$ nanograms and 1 kilogram = 1000 grams.

A space probe is sent on a mission to a distant exoplanet. The distance to the exoplanet is 1.2 billion kilometres. The probe travels at a constant speed of 50,000 kilometres per hour.

(a) Calculate the time, in days, it will take for the probe to reach the exoplanet.

The probe is designed to collect interstellar dust particles. On its journey, it collects an average of $2.5 \times 10^4$ particles for every kilometre it travels.

(b) Find the total number of interstellar dust particles the probe will collect on its journey to the exoplanet. Give your answer in the form $a \times 10^{k}$ where $1 \le a < 10$ and $k \in \mathbb{Z}$.

(a) The efficiency of a new solar panel design, $E$, is modeled by the formula:

$E = \frac{ \sin \alpha + \cos \beta }{ \sqrt{ R^2 + X } }$

where $\alpha$ and $\beta$ are angles in degrees, $R$ is a resistance value, and $X$ is a reactance value.

Given $\alpha = 25^{\circ}$, $\beta = 70^{\circ}$, $R = 15.3$ ohms, and $X = 48.7$ ohms, calculate the value of $E$. Write down your full calculator display.

(b) Write your answer to part (a):

(i) correct to three decimal places

(ii) correct to four significant figures.

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