Exercises: Standard form

The video of a well-known Mexican streamer has reached $45,200,000$ views in just one week. Express this number of views in scientific notation and state its mantissa and its order of magnitude.

In the biology laboratory, a group of students observes that the diameter of a red blood cell is approximately $7.5 \times 10^{-6}$ meters. Express this measurement in decimal form (ordinary number notation) and explain in which direction the decimal point moves and by how many places.

State which of the following numbers are correctly written in standard form. If any are not, briefly explain why:

1. a) $0.45 \times 10^{8}$
2. b) $1.02 \times 10^{-5}$
3. c) $12.5 \times 10^{3}$
4. d) $9.99 \times 10^{12}$

A technology company is comparing the size of three electronic components. The sizes are: $A = 3.2 \times 10^{-4}$ mm, $B = 1.5 \times 10^{-3}$ mm and $C = 8.9 \times 10^{-4}$ mm. Order the components from smallest to largest based on their exponents and mantissas.

The mean distance from the Earth to the Sun is approximately $1.5 \times 10^{8}$ km. If light travels at a speed of $3 \times 10^{5}$ km/s, calculate the time it takes for sunlight to reach the Earth, carrying out the operation directly in standard form (remember that $t = \frac{d}{v}$).

A data server stores music files. Each song occupies an average of $4.5 \times 10^{6}$ bytes. If the server has $2 \times 10^{4}$ songs stored, how many total bytes of music are on the server? Express the final result in standard form.

Determine the value of $x$ in each of the following equations so that equivalence in standard form is maintained:

1. a) $345,000 = 3.45 \times 10^{x}$
2. b) $0.00007 = 7 \times 10^{x}$
3. c) $12.5 \times 10^{4} = 1.25 \times 10^{x}$

Look at the following diagram representing the movement of the decimal point to convert the number $0.000042$ to standard form. Complete the boxes with the value of the coefficient (mantissa) and the corresponding exponent.

In a data processing center of a well-known social network, it is estimated that $2.5 \times 10^6$ megabytes (MB) of new information are generated every second.

1. a) Express in scientific notation how many gigabytes (GB) are generated in a full day (assume $1 \text{ GB} = 10^3 \text{ MB}$).

1. b) If each storage hard drive has a capacity of $1.2 \times 10^{13}$ bytes, how many hard drives would be needed to store the information of a leap year? (assume $1 \text{ MB} = 10^6 \text{ bytes}$). Give the result to three significant figures.

The mass of a suspended dust particle is approximately $7.25 \times 10^{-10}$ kg. After one week of operation, an air purifier has captured a total of $4.35 \times 10^{-2}$ grams of these particles.

1. a) Calculate the total number of particles captured by the purifier.

1. b) If the purifier filter must be changed when it reaches an accumulated mass of $1.5 \text{ decigramos}$, how many more weeks can it operate before requiring maintenance? Express the calculations and the final result in standard form.

A group of researchers is analysing R&D investment by three world powers. The first invests $4.2 \times 10^{11} \$$, the second $1.85 \times 10^{12} \$$ and the third $9.7 \times 10^{10} \$$.

1. a) Calculate the total investment of the three powers combined, expressing the result in standard form.

1. b) What percentage of the total is represented by the investment of the power that contributes the least money? Round the percentage to two decimal places.

In microelectronics, the thickness of an insulating layer in a processor is $4.5$ nanometers ($1 \text{ nm} = 10^{-9} \text{ m}$). If the processor has a total surface area of $1.2 \text{ cm}^2$:

1. a) Express the thickness of the layer in meters using scientific notation.

1. b) Calculate the total volume of insulating material used in that processor in cubic meters ($m^3$). Take into account that the volume is the product of the surface area and the thickness.

Carry out the following combined operations and express the result in standard form, rounding to three significant figures:

A red blood cell has an approximately circular shape with a diameter of $7.5 \times 10^{-4} \text{ cm}$.

1. a) Calculate the circumference of the cell in meters. (Use $\pi \approx 3.1416$).

1. b) If we could line up end to end all the red blood cells of an adult (approximately $2.5 \times 10^{13}$ cells), what distance in kilometers would they cover? Express the result in scientific notation.

The mean distance from the Earth to the Sun is $1.496 \times 10^8 \text{ km}$. The speed of light is $3 \times 10^8 \text{ m/s}$.

1. a) Calculate how long, in minutes and seconds, it takes for sunlight to reach the Earth.

1. b) If a spacecraft travels at a constant speed of $5.5 \times 10^4 \text{ km/h}$, how many days would it take to complete the round trip?

The mass of a laboratory object has been measured, giving a value of $3.45 \times 10^{-3}$ grams, although the true value is $3.458 \times 10^{-3}$ grams.

1. a) Calculate the absolute error made in the measurement.

1. b) Calculate the relative error and express it as a percentage. Is it considered a precise measurement if the acceptable margin of error is $0.5\%$? Justify your answer.

In a nanotechnology laboratory, the mass of a new synthetic particle is being analysed, estimated at $m = 4.25 \times 10^{-22}$ grams. However, the measuring instrument has an absolute error bound of $5 \times 10^{-25}$ grams. On the other hand, an astronomer measures the mass of a nearby asteroid as $M = 7.8 \times 10^{15}$ kilograms with a relative error of $0.01\%$.

Carry out the following tasks, expressing all results in standard form:

1. a) Determine the interval of possible values for the mass of the particle $m$ using neighbourhood notation $E(a, r)$.

1. b) Calculate the relative error bound made in measuring the particle and compare it with the relative error of the asteroid. Give reasons to justify which measurement is more precise.

1. c) If $1.5 \times 10^{24}$ identical particles were combined, what would be the resulting total mass? Consider both the estimated value and the accumulated absolute error.

Calculate the value of the following expression, simplifying the result as far as possible, and express it in standard form to three significant figures:

Next, bound the absolute error made when rounding the value of $L$ to the first decimal place of its mantissa.

A social network data center processes approximately $2.5 \times 10^6$ queries per second. Each query requires the movement of $1.2 \times 10^4$ bytes of information.

1. a) Calculate how many terabytes ($1 \text{ TB} = 10^{12} \text{ bytes}$) of information the center processes in a leap year.

1. b) If the center plans to increase its processing capacity by $15\%$ annually compounded, how much information (in bytes) will it process per second in a decade? Express the result in scientific notation.

The golden ratio, $\phi = \frac{1 + \sqrt{5}}{2}$, is present in the design of many modern mobile application interfaces to achieve visual harmony. Consider a smartphone screen following this ratio, where the width measures $w = 6.8 \times 10^{-2}$ meters.

1. a) Calculate the area of the screen in square millimeters ($mm^2$) knowing that the ratio between the height ($h$) and the width ($w$) is exactly $\phi$. Use the radical expression of $\phi$ for the calculation and approximate the final result in scientific notation to two decimal places.

1. b) If an error of $1 \times 10^{-4}$ meters in width occurs when manufacturing the glass (the height remaining unchanged), how does this affect the relative error of the area?

In a global reforestation campaign, it is estimated that each planted tree absorbs an average of $2.2 \times 10^1$ kg of $CO_2$ per year. The organising platform claims to have offset $4.5 \times 10^8$ metric tons of $CO_2$ over the last 5 years ($1 \text{ tonelada} = 10^3 \text{ kg}$).

1. a) Assuming that the number of trees planted has been constant and that absorption is cumulative (trees planted in the first year absorb over all five years, those planted in the second year over four years, etc.), calculate how many trees were planted each year.

1. b) If there is an error in the absorption measurement of $\pm 5\%$ per tree, calculate the interval for the absolute error of the total number of metric tons offset.

Represent on the number line the set of numbers satisfying the following "industrial tolerance" condition for a precision part:

1. a) Express the set in interval form and as an inequality without modulus.

1. b) Is the number $0.00435$ an acceptable value for this part? Justify your answer by checking whether it belongs to the set.

You can do it on paper and send a photo with your answer.

The world population is projected to reach $9.7 \times 10^9$ people in the year 2050. If the total land surface area is approximately $1.48 \times 10^8 \text{ km}^2$, of which only $25\%$ is habitable:

1. a) Calculate the population density (inhabitants per $km^2$) in the habitable areas for the year 2050.

1. b) If each person consumes an average of $1.3 \times 10^6$ kilocalories per year, how much total food energy (in kcal) will humanity need in a decade (10 years), assuming the population remains constant at the 2050 figure? Give the result in standard form.

Given the following diagram representing a network of underground fiber-optic cables, the distance between inspection hatches $A$ and $B$ is given by the expression $d = \sqrt{5 \times 10^{-4}} + \sqrt{2 \times 10^{-5}}$ kilometers.

1. a) Simplify the expression for distance $d$ by taking factors out of the radicals and performing the operations to leave a single radical (or the simplest possible sum).

1. b) If a repair requires a cable whose length is exactly three times $d$, calculate this length and express it in meters using scientific notation to four significant figures.