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Difficulty: MediumExponents, Roots, and Scientific Notation

A certain region has a total land area of 2.5×1042.5 \times 10^4 square miles and a population of 8.0×1058.0 \times 10^5 people. What is the average population density of this region, in people per square mile?

  1. 3.2×1013.2 \times 10^1Answer
  2. B
    3.125×10−23.125 \times 10^{-2}
  3. C
    2.0×10102.0 \times 10^{10}
  4. D
    3.2×1093.2 \times 10^9
  5. E
    3.2×10203.2 \times 10^{20}

Answer

The population density is 3.2×1013.2 \times 10^1 people per square mile.
To find the average population density, we divide the population by the land area: 8.0×1052.5×104\frac{8.0 \times 10^5}{2.5 \times 10^4}. Dividing the coefficients gives 8.02.5=3.2\frac{8.0}{2.5} = 3.2. Subtracting the exponents of the powers of 10 gives 105−4=10110^{5-4} = 10^1. Combining these results yields the correct density of 3.2×1013.2 \times 10^1 people per square mile.

Step-by-Step Solution

1
Set up the ratio for population density.
Density=8.0×105 people2.5×104 square miles\text{Density} = \frac{8.0 \times 10^5 \text{ people}}{2.5 \times 10^4 \text{ square miles}}
Population density is defined as population divided by land area.
2
Divide the numerical coefficients.
8.02.5=3.2\frac{8.0}{2.5} = 3.2
To divide numbers in scientific notation, first divide their lead coefficients.
3
Divide the powers of 10 by subtracting the exponent of the denominator from the exponent of the numerator.
105104=105−4=101\frac{10^5}{10^4} = 10^{5-4} = 10^1
According to the quotient rule of exponents, xaxb=xa−b\frac{x^a}{x^b} = x^{a-b}.
4
Combine the simplified coefficient and power of 10.
3.2×1013.2 \times 10^1
This is the standard scientific notation form of the final answer.

Key Concept

Operations with numbers in scientific notation
Estimated Time:1m 0s
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Topics

Exponents, Roots, and Scientific Notation

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