Scientific Notation in a District's Cloud Archive
A school district scans old student records into a shared cloud archive. Each of the district's 24 schools uploads 3.6 × 10^5 kilobytes of scans per month. The archive currently has 2 terabytes of unused capacity, and the technology coordinator plans using 1 terabyte = 10^9 kilobytes, ignoring the small difference between decimal and binary prefixes. A student intern multiplies 24 by 3.6 × 10^5, writes the product as 8.64 × 10^5 KB, and tells the board the archive will last “well over 200 years.” The coordinator is skeptical, because the drive must also absorb growth: average scan file size has been rising about 15% per year while the number of schools stays fixed. She needs a defensible statement for the board about whether a 10-year capacity plan is actually safe.
1. Check the intern's monthly total and the 200-year claim. What is the correct monthly total in scientific notation, and about how many years of storage remain at the current upload rate?
Read the worked answer
Multiply the coefficient by the number of schools: 24 × 3.6 = 86.4, not 8.64. So the monthly upload is 86.4 × 10^5 = 8.64 × 10^6 KB.
The intern made a place-value error of one power of ten - a factor of 10 - which is exactly the kind of slip that turns 19 years into "200 years."
Capacity in the coordinator's units: 2 × 10^9 KB.
Months of storage = (2 × 10^9) ÷ (8.64 × 10^6) = (2 ÷ 8.64) × 10^3 ≈ 231.5 months ≈ 19.3 years.
So at the current rate the drive holds roughly 19 years, and the intern's claim is too optimistic by about a factor of ten. Even the corrected figure assumes nothing grows, which the coordinator already knows is false.
2. The board asks whether a 10-year plan is safe if average file size grows 15% per year with 24 schools. Using 1.15^10 ≈ 4.05 and 1.15^8 ≈ 3.06, estimate cumulative uploads over 10 years and over 8 years, then decide.
Read the worked answer
First convert to a yearly figure: 12 × 8.64 × 10^6 KB ≈ 1.04 × 10^8 KB in year 1.
With 15% annual growth the yearly totals form a geometric series, so the 10-year sum is 1.04 × 10^8 × (1.15^10 − 1) ÷ 0.15.
- 1.15^10 − 1 ≈ 4.05 − 1 = 3.05.
- 3.05 ÷ 0.15 ≈ 20.3.
- 1.04 × 10^8 × 20.3 ≈ 2.11 × 10^9 KB.
That exceeds the 2 × 10^9 KB capacity. Checking 8 years: 3.06 − 1 = 2.06; 2.06 ÷ 0.15 ≈ 13.7; 1.04 × 10^8 × 13.7 ≈ 1.42 × 10^9 KB, which fits with room to spare.
Conclusion for the board: growth moves the wall from about year 19 to roughly year 9. A 10-year plan is not safe on the current drive; the district needs added capacity or a plan to archive older scans elsewhere.
3. The intern later writes (3.6 × 10^5)^2 = 7.2 × 10^10. Identify the error and give the correct value in scientific notation.
Read the worked answer
Squaring a number in scientific notation means squaring the coefficient and multiplying the exponent by 2 - not doubling either one.
(3.6 × 10^5)^2 = 3.6^2 × (10^5)^2 = 12.96 × 10^10.
Convert to proper scientific notation by moving the decimal one place left and adding 1 to the exponent: 1.296 × 10^11.
The intern doubled the coefficient (3.6 → 7.2) and doubled the exponent (5 → 10) instead of squaring. A magnitude check catches this: 3.6 × 10^5 is a bit under 4 × 10^5, so its square should be a bit under 16 × 10^10 = 1.6 × 10^11. The answer 1.296 × 10^11 is consistent; 7.2 × 10^10 is too small.