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Difficulty: HardDecimals and Scientific Notation

A chemical laboratory starts with a liquid solution having a total volume of 3.2×10−43.2 \times 10^{-4} cubic meters. The solution is partitioned equally into 400400 identical micro-vials. Subsequently, heat treatment causes the volume of liquid in each micro-vial to evaporate, reducing its volume by 75%75\%. What is the final volume of liquid remaining in a single micro-vial, expressed in scientific notation?

  1. 2.0×10−72.0 \times 10^{-7} cubic metersAnswer
  2. B
    8.0×10−78.0 \times 10^{-7} cubic meters
  3. C
    6.0×10−76.0 \times 10^{-7} cubic meters
  4. D
    2.0×10−52.0 \times 10^{-5} cubic meters
  5. E
    2.0×10−62.0 \times 10^{-6} cubic meters

Answer

2.0×10−72.0 \times 10^{-7} cubic meters
Dividing the initial total volume of 3.2×10−43.2 \times 10^{-4} cubic meters by 400400 (4×1024 \times 10^2) gives an initial volume of 8.0×10−78.0 \times 10^{-7} cubic meters per micro-vial. Reducing this volume by 75%75\% means 25%25\% of the liquid remains. Multiplying 8.0×10−78.0 \times 10^{-7} by 0.250.25 yields 2.0×10−72.0 \times 10^{-7} cubic meters.

Step-by-Step Solution

1
Express the number of micro-vials in scientific notation.
400=4×102400 = 4 \times 10^2
Converting into scientific notation simplifies division involving powers of 10.
2
Calculate the initial volume per micro-vial before evaporation.
3.2×10−44×102=(3.24)×10−4−2=0.8×10−6=8.0×10−7\frac{3.2 \times 10^{-4}}{4 \times 10^2} = \left(\frac{3.2}{4}\right) \times 10^{-4 - 2} = 0.8 \times 10^{-6} = 8.0 \times 10^{-7} cubic meters
Equal partitioning requires dividing the total volume by the total number of containers.
3
Determine the remaining fraction of liquid after a 75%75\% volume reduction.
Remaining fraction =100%−75%=25%=0.25= 100\% - 75\% = 25\% = 0.25
A reduction by 75%75\% leaves 25%25\% of the liquid volume in the vial.
4
Multiply the volume per micro-vial by the remaining fraction.
8.0×10−7×0.25=2.0×10−78.0 \times 10^{-7} \times 0.25 = 2.0 \times 10^{-7} cubic meters
Applying the remaining fraction yields the final volume in standard scientific notation format a×10na \times 10^n where 1≤a<101 \le a < 10.

Key Concept

Operations with Decimals, Place Value, and Scientific Notation
Estimated Time:2m 0s
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Topics

Decimals and Scientific Notation

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