Question 9 of 28intermediate🔧 ApplyNumerical3 marks

A dairy plans to produce 8.5 billion packets of milk in a year. They want a unique ID (identifier) code for each packet. If they choose to use the digits 0–9, how many digits should the code consist of?

Correct Answer

Given: 8.5 billion = 8.5×1098.5 \times 10^9

Formula: An n-digit code using digits 0-9 can create 10n10^n unique codes

Requirement: We need 10n≥8.5×10910^n \geq 8.5 \times 10^9

Calculation: 109=110^9 = 1 billion codes (not enough) 1010=1010^{10} = 10 billion codes (sufficient)

Answer: The code should consist of 10 digits.

Exercise: Figure it Out (Section 2) | Q: 9 | (Page Page 27)
For More Understanding

Explanation

8.5 billion = 8.5×1098.5 \times 10^9. An n-digit code using digits 0-9 can create 10n10^n unique codes. We need 10n≥8.5×10910^n \geq 8.5 \times 10^9. 109=110^9 = 1 billion codes, which is not enough. 1010=1010^{10} = 10 billion codes, which is sufficient. So the code needs 10 digits.

Solution Steps

  1. Step 1: Convert 8.5 billion to scientific notation: 8.5×1098.5 \times 10^9

  2. Step 2: An n-digit code using digits 0-9 can create 10n10^n unique codes

  3. Step 3: We need 10n≥8.5×10910^n \geq 8.5 \times 10^9

  4. Step 4: 109=110^9 = 1 billion, which is less than 8.5 billion

  5. Step 5: 1010=1010^{10} = 10 billion, which is greater than 8.5 billion

  6. Step 6: Therefore, 10 digits are needed

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