Scientific notation AI HL P1
Scientific Notation, Spherical Modelling & Percentage Error
IB Mathematics HL • Scientific Notation • Modelling with Spheres • Estimation
⭐ Key Concepts
- Scientific notation: numbers written as
where 
. - Volume of a sphere
![\[V=\frac{4}{3}\pi r^3\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-f7dba31d09dd5d64ca5a4846f470f586_l3.png?resize=86%2C41&ssl=1 "Rendered by QuickLaTeX.com")
- If diameter
is known:
![\[V=\frac{\pi}{6}d^3\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-28a3a985a42d939f520c160790ee8e34_l3.png?resize=77%2C37&ssl=1 "Rendered by QuickLaTeX.com")
- Surface area of a sphere
![\[S=4\pi r^2\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-24d8a8ba2fa1fd41754145c60e353de4_l3.png?resize=78%2C18&ssl=1 "Rendered by QuickLaTeX.com")
- Percentage error
![\[\text{Error}=\left|\frac{\text{estimate}-\text{actual}}{\text{actual}}\right|\times100\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-da5284e751def8c1cb66e75dfc340843_l3.png?resize=297%2C49&ssl=1 "Rendered by QuickLaTeX.com")
📘 IB Command Terms
- Estimate – obtain an approximate value using a model.
- Find – calculate an exact or numerical answer.
- Determine – obtain an answer with reasoning.
- Calculate – obtain a numerical result using a formula.
🧮 GDC Calculator Tips
- Use scientific notation mode when working with powers of ten.
- On most GDCs, use the EE or EXP key for
. - Enter expressions like
![\[(1.75E{-10})^3\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-f415e7e38f5b679a242caeba24e3c18d_l3.png?resize=110%2C23&ssl=1 "Rendered by QuickLaTeX.com")
- Store intermediate answers to avoid rounding errors.
📌 Example 1 — Uranium Nucleus
A uranium nucleus has diameter 
m.
(a) Write the diameter in scientific notation.
(b) Estimate the volume of the nucleus assuming it is spherical.
Solution
(a)
![\[d=1.0\times10^{-15}\text{ m}\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-a37c6ce8c779703144857d63daf11906_l3.png?resize=159%2C18&ssl=1 "Rendered by QuickLaTeX.com")
📊 Marks: M1 A1
(b)
![\[V=\frac{\pi}{6}(1.0\times10^{-15})^3\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-8234e307088cfc1709122627ce969979_l3.png?resize=181%2C37&ssl=1 "Rendered by QuickLaTeX.com")
![\[V=\frac{\pi}{6}\times10^{-45}\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-90ea4f0dcd0a7756a0b9f5509b672525_l3.png?resize=130%2C37&ssl=1 "Rendered by QuickLaTeX.com")
![\[V\approx5.24\times10^{-46}\text{ m}^3\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-9af77582ef5e262c3987bec30d8f0db4_l3.png?resize=181%2C18&ssl=1 "Rendered by QuickLaTeX.com")
📊 Marks: M1 M1 A1 A1
📌 Example 2 — Lead Atom Model
The diameter of a lead atom is 
nm.
(a) Convert the diameter into metres.
(b) Estimate the volume of the atom.
(c) Estimate how many atoms fit inside 
.
Solution
(a)
![\[0.175\text{ nm}=1.75\times10^{-10}\text{ m}\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-b49e92762f97482a577ca256961b7931_l3.png?resize=239%2C18&ssl=1 "Rendered by QuickLaTeX.com")
📊 Marks: M1 A1
(b)
![\[V=\frac{\pi}{6}(1.75\times10^{-10})^3\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-0873aa0ee85dbc00db133b040571e921_l3.png?resize=191%2C37&ssl=1 "Rendered by QuickLaTeX.com")
![\[V\approx2.81\times10^{-30}\text{ m}^3\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-d40b3e310eb51c64bac27c6088979075_l3.png?resize=181%2C18&ssl=1 "Rendered by QuickLaTeX.com")
📊 Marks: M1 M1 A1
(c)
![\[N=\frac{1.0\times10^{-6}}{2.81\times10^{-30}}\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-389799f720884e311109d947e722e9ad_l3.png?resize=157%2C45&ssl=1 "Rendered by QuickLaTeX.com")
![\[N\approx3.56\times10^{23}\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-f9dc8b7175f810ff867f876b8fc99ea9_l3.png?resize=141%2C18&ssl=1 "Rendered by QuickLaTeX.com")
📊 Marks: M1 A1
⚠ Common Mistakes
- Forgetting to convert nm or cm³ into metres.
- Cubing the number but not cubing the power of ten.
- Using
when the formula requires 
. - Rounding too early during calculations.
IB HL Paper 1 Style Exam Questions
Question 1
A spherical asteroid has radius 
km.
(a) Write 
quadrillion euros in the form .
(b) Estimate the asteroid’s volume.
(c) The actual volume is 
km³.
Calculate the percentage error.
🧪 HL Challenge Problems
Challenge 1
A virus particle is approximated by a sphere with diameter 
m.
(a) Express the diameter in scientific notation.
(b) Estimate the volume of the virus.
(c) If a droplet has volume 
m³, estimate how many viruses could fit inside.
Challenge 2
A spherical planet has radius 
km.
(a) Find the surface area of the planet.
(b) If oceans cover 
km², calculate the percentage covered by water.
(c) Compare this with Earth’s water coverage.
✅ Self-Practice (IB Paper 1 Style)
Question 1
A nucleus has diameter 
m.
(a) Write the diameter in scientific notation.
(b) Estimate the volume.
(c) Express the answer to 3 significant figures.
Question 2
An atom has diameter 
nm.
(a) Convert the diameter into metres.
(b) Estimate the volume.
(c) Estimate how many atoms fit in 
mm³.
Show Answers
1
![\[V=\frac{\pi}{6}(3\times10^{-15})^3\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-4f7ca51b8c370337610d1b4f9b9e09ad_l3.png?resize=165%2C37&ssl=1 "Rendered by QuickLaTeX.com")
![\[V\approx1.41\times10^{-44}\text{ m}^3\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-787a449441d84eaecd09dd31c57b1e8b_l3.png?resize=181%2C18&ssl=1 "Rendered by QuickLaTeX.com")
2
![\[d=3.00\times10^{-10}\text{ m}\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-0c44ffde5d603f294cb5f2e1d0865317_l3.png?resize=169%2C18&ssl=1 "Rendered by QuickLaTeX.com")
![\[V\approx1.41\times10^{-29}\text{ m}^3\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-e5dbbecdf44fa715648102c8fdcef02c_l3.png?resize=181%2C18&ssl=1 "Rendered by QuickLaTeX.com")