Scientific notation AI SL Paper 1
Scientific Notation, Volume & Percentage Error
IB Mathematics • Scientific Notation • Modelling with Spheres • Percentage Error
⭐ Key Concepts
- Scientific notation: numbers written as
where 
. - Large quantities in astronomy, physics, and finance are often written using powers of 10.
- 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")
- 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")
⚠ Common Mistakes
- Writing scientific notation with
(e.g. 
instead of 
). - Forgetting that
. - Not cubing the radius when using the sphere formula.
- Incorrectly subtracting values with powers of ten.
- Using estimated value in the denominator when calculating percentage error.
📘 Exam Tips
- Always convert the coefficient into the range .
- Factor powers of ten when subtracting numbers written in scientific notation.
- Keep values in standard form until the final step to avoid rounding errors.
- Write the formula you are using — this earns method marks.
- Round answers only at the final stage (usually to 3 s.f.).
Question 1
A rare metallic asteroid contains platinum deposits valued at
9875 quadrillion euros.
![\[1\text{ quadrillion}=10^{15}\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-3d9ccfec87e6020422aaaf6de7eeeed5_l3.png?resize=171%2C22&ssl=1 "Rendered by QuickLaTeX.com")
(a) Express the value in scientific notation.
(b) Model the asteroid as a sphere of radius 
km.
Find its volume.
(c) The actual volume is 
km
.
Find the percentage error of the estimate.
Solution
(a) Scientific notation
![\[9875\times10^{15}\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-140842c45f789b4375bc1294586ad2e5_l3.png?resize=99%2C18&ssl=1 "Rendered by QuickLaTeX.com")
Write
![\[9875=9.875\times10^3\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-a41c18034b16b264d14d0d3d17adaf10_l3.png?resize=164%2C18&ssl=1 "Rendered by QuickLaTeX.com")
Thus
![\[(9.875\times10^3)\times10^{15}\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-48f5c392095f9b0cefa3af8c07f387fd_l3.png?resize=173%2C23&ssl=1 "Rendered by QuickLaTeX.com")
![\[=9.875\times10^{18}\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-5dfba0a8abd3dcd370149f41757a0ecf_l3.png?resize=126%2C18&ssl=1 "Rendered by QuickLaTeX.com")
![\[\boxed{9.875\times10^{18}}\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-391d0ec0d479b937bc5c0ec0f0b37115_l3.png?resize=119%2C32&ssl=1 "Rendered by QuickLaTeX.com")
(b) Volume of the asteroid
![\[r=128\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-ed182a774c4936ccd80c5499a37ab8aa_l3.png?resize=67%2C14&ssl=1 "Rendered by QuickLaTeX.com")
![\[128^3=2\,097\,152\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-e7e974116cf9a6ae16394360a6cbbbc1_l3.png?resize=140%2C18&ssl=1 "Rendered by QuickLaTeX.com")
![\[V=\frac{4}{3}\pi(2\,097\,152)\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-009987f8665095a352814370db30b338_l3.png?resize=158%2C41&ssl=1 "Rendered by QuickLaTeX.com")
![\[V\approx8.79\times10^6\text{ km}^3\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-b199daa7b32a1e087ff8ce633282f885_l3.png?resize=173%2C18&ssl=1 "Rendered by QuickLaTeX.com")
![\[\boxed{8.79\times10^6}\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-3e2bf16367f4c6fe853e43c929678cd2_l3.png?resize=102%2C32&ssl=1 "Rendered by QuickLaTeX.com")
(c) Percentage error
![\[\left|\frac{8.79\times10^6-7.385\times10^6}{7.385\times10^6}\right|\times100\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-c247859d708776dff1cc91109394f336_l3.png?resize=282%2C50&ssl=1 "Rendered by QuickLaTeX.com")
Factor 
![\[\left|\frac{8.79-7.385}{7.385}\right|\times100\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-536cd14cee1d8767571f84d06ded701b_l3.png?resize=175%2C49&ssl=1 "Rendered by QuickLaTeX.com")
![\[\approx19.0\%\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-6bf76bfc2a7cf19d2c1e55b914525945_l3.png?resize=72%2C16&ssl=1 "Rendered by QuickLaTeX.com")
![\[\boxed{19.0\%}\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-a4667450f9e2c878cdd26398c3e6fad8_l3.png?resize=65%2C29&ssl=1 "Rendered by QuickLaTeX.com")
Question 2
An exoplanet fragment is valued at 12,345 trillion euros.
![\[1\text{ trillion}=10^{12}\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-803f1b213223f4b9037a9c756a4832e7_l3.png?resize=136%2C18&ssl=1 "Rendered by QuickLaTeX.com")
(a) Express the value in scientific notation.
(b) Model the fragment as a sphere of radius 
km and find the volume.
(c) The actual volume is 
km.
Calculate the percentage error.
Solution
(a)
![\[12345\times10^{12}\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-eb0ab5b8a516414af7cff0c152cc5f84_l3.png?resize=108%2C18&ssl=1 "Rendered by QuickLaTeX.com")
![\[12345=1.2345\times10^4\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-a049f42dc78daa6a57b21fc3d458b3ab_l3.png?resize=183%2C18&ssl=1 "Rendered by QuickLaTeX.com")
![\[=1.2345\times10^{16}\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-5f1ca2372fe60620cfe4a7aa723d4051_l3.png?resize=136%2C18&ssl=1 "Rendered by QuickLaTeX.com")
To 3 s.f.
![\[\boxed{1.23\times10^{16}}\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-ad3b36c91637d9947e04d344a2d668e7_l3.png?resize=109%2C32&ssl=1 "Rendered by QuickLaTeX.com")
(b)
![\[V=\frac{4}{3}\pi(2100)^3\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-6b07036877e6c6ad1b6c7a20e4af4591_l3.png?resize=132%2C41&ssl=1 "Rendered by QuickLaTeX.com")
![\[2100^3=9.261\times10^9\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-0b756222222aaffbc7d1076eaa4c5782_l3.png?resize=174%2C18&ssl=1 "Rendered by QuickLaTeX.com")
![\[V\approx3.88\times10^{10}\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-ade1d35131d34f37eb40b9e20674de02_l3.png?resize=138%2C18&ssl=1 "Rendered by QuickLaTeX.com")
![\[\boxed{3.88\times10^{10}}\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-26bc8baef90214e6ed935a02a0b138d8_l3.png?resize=109%2C32&ssl=1 "Rendered by QuickLaTeX.com")
(c)
![\[\left|\frac{3.88\times10^{10}-4.71\times10^{10}}{4.71\times10^{10}}\right|\times100\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-abc17356cc91b5b2a5e712ed56f086f6_l3.png?resize=287%2C50&ssl=1 "Rendered by QuickLaTeX.com")
![\[\approx17.6\%\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-489cac5084a2014119124daf5d0d65f3_l3.png?resize=72%2C16&ssl=1 "Rendered by QuickLaTeX.com")
![\[\boxed{17.6\%}\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-63eac2f9f04a53b7223407a0995e9ec5_l3.png?resize=65%2C29&ssl=1 "Rendered by QuickLaTeX.com")
🧪 Challenge Problems
-
The mass of Earth is approximately
kg.
Express this in kilograms written without scientific notation. -
A spherical asteroid has radius
km.
Calculate its volume in scientific notation. -
An estimate of a star’s mass is
kg but the true value is
kg.
Find the percentage error.
Self-Practice Questions
- Write 4250 quadrillion in scientific notation.
- Find the volume of a sphere with radius
km. - The estimated volume of an asteroid is
km and the exact value is 
km.
Find the percentage error. - Write 678 trillion in scientific notation.
- Write 9600 billion in scientific notation.
Click to Reveal Solutions
![\[4250\times10^{15}=4.25\times10^{18}\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-50e74318589b157af42bbf00005c1643_l3.png?resize=224%2C18&ssl=1 "Rendered by QuickLaTeX.com")
![\[V=\frac{4}{3}\pi(85^3)\approx2.57\times10^6\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-51307c448c67e6f2842df27f7ed60ff7_l3.png?resize=227%2C41&ssl=1 "Rendered by QuickLaTeX.com")
![\[\text{Error}=16.8\%\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-7ed95d06b205123e8dc3d048528d8734_l3.png?resize=126%2C16&ssl=1 "Rendered by QuickLaTeX.com")
![\[678\times10^{12}=6.78\times10^{14}\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-0152e9fc80a0be172399892016186d16_l3.png?resize=214%2C18&ssl=1 "Rendered by QuickLaTeX.com")
![\[9600\times10^9=9.60\times10^{12}\]](https://i0.wp.com/alphyschool.org/wp-content/ql-cache/quicklatex.com-01cd8c2d678ed5d3169bbb553bc334d5_l3.png?resize=215%2C18&ssl=1 "Rendered by QuickLaTeX.com")