Examkin

Question

Difficulty: MediumScientific Notation

A city water purification system processes 8.5×1058.5 \times 10^5 liters of water per day. A specialized chemical treatment requires 4.0×10−44.0 \times 10^{-4} grams of a purifying agent per liter of water. What is the total mass, in grams, of the purifying agent needed each day? Express the answer in proper scientific notation.

  1. A
    3.4×1003.4 \times 10^0
  2. B
    3.4×10−193.4 \times 10^{-19}
  3. 3.4×1023.4 \times 10^2Answer
  4. D
    3.4×10103.4 \times 10^{10}

Answer

The correct answer is 3.4×1023.4 \times 10^2.
The total mass is calculated by multiplying the daily volume by the required agent per liter: (8.5×105)×(4.0×10−4)(8.5 \times 10^5) \times (4.0 \times 10^{-4}). Multiplying the coefficients gives 3434, and adding the exponents yields 10110^1. The intermediate result, 34×10134 \times 10^1, normalizes to 3.4×1023.4 \times 10^2 by shifting the decimal one place to the left and adding 11 to the exponent.

Step-by-Step Solution

1
Set up the multiplication expression to find the total mass.
(8.5×105)×(4.0×10−4)(8.5 \times 10^5) \times (4.0 \times 10^{-4})
The total mass is the product of the total daily volume of water and the mass of the purifying agent required per liter.
2
Group the coefficients and the powers of ten.
(8.5×4.0)×(105×10−4)(8.5 \times 4.0) \times (10^5 \times 10^{-4})
The commutative and associative properties of multiplication allow grouping similar numerical components.
3
Multiply the coefficients and add the exponents.
34×10134 \times 10^1
8.5×4.0=348.5 \times 4.0 = 34, and following the product rule of exponents, 105×10−4=105+(−4)=10110^5 \times 10^{-4} = 10^{5 + (-4)} = 10^1.
4
Normalize the result into proper scientific notation.
3.4×1023.4 \times 10^2
The coefficient 3434 must be converted to 3.43.4 (a number between 1 and 10). Because the decimal point is moved one place to the left, the exponent is increased by 1.

Key Concept

Multiplying Numbers in Scientific Notation and Normalization
Rate this question

Topics

Scientific Notation

More questions from this topic