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Difficulty: MediumExponents, Roots, and Scientific Notation

The volume of a spherical cell is given by the formula V=43πr3V = \frac{4}{3}\pi r^3, where rr is the radius of the cell. If a certain spherical cell has a radius of r=3.0×10−6r = 3.0 \times 10^{-6} meters, what is its volume in cubic meters, written in scientific notation?

  1. A
    1.2π×10−171.2\pi \times 10^{-17}
  2. B
    1.2π×10−111.2\pi \times 10^{-11}
  3. C
    3.6π×10−23.6\pi \times 10^{-2}
  4. 3.6π×10−173.6\pi \times 10^{-17}Answer
  5. E
    6.4π×10−176.4\pi \times 10^{-17}

Answer

3.6π×10−173.6\pi \times 10^{-17}
To find the volume, substitute the radius r=3.0×10−6r = 3.0 \times 10^{-6} into the formula V=43πr3V = \frac{4}{3}\pi r^3. First, cube the radius: (3.0×10−6)3=27×10−18(3.0 \times 10^{-6})^3 = 27 \times 10^{-18}. Next, multiply this value by 43π\frac{4}{3}\pi to get 36π×10−1836\pi \times 10^{-18}. Finally, convert this value to standard scientific notation by rewriting it as 3.6π×10−173.6\pi \times 10^{-17}.

Step-by-Step Solution

1
Substitute the radius r=3.0×10−6r = 3.0 \times 10^{-6} meters into the sphere volume formula V=43πr3V = \frac{4}{3}\pi r^3.
V=43π(3.0×10−6)3V = \frac{4}{3}\pi (3.0 \times 10^{-6})^3
This sets up the expression for calculating the volume.
2
Cube the term inside the parentheses using the power of a product and power of a power rules.
(3.0×10−6)3=3.03×(10−6)3=27×10−18(3.0 \times 10^{-6})^3 = 3.0^3 \times (10^{-6})^3 = 27 \times 10^{-18}
This simplifies the radius term before performing the remaining operations.
3
Multiply the simplified term by 43π\frac{4}{3}\pi.
V=43π×27×10−18=36π×10−18V = \frac{4}{3}\pi \times 27 \times 10^{-18} = 36\pi \times 10^{-18}
This performs the fractional multiplication to find the volume.
4
Convert the volume into standard scientific notation by rewriting 3636 as 3.6×1013.6 \times 10^1 and combining the exponents.
36π×10−18=3.6π×10−1736\pi \times 10^{-18} = 3.6\pi \times 10^{-17}
Standard scientific notation requires the coefficient to be between 1 and 10.

Key Concept

Applying laws of exponents and scientific notation rules to evaluate formulas.
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Topics

Exponents, Roots, and Scientific Notation

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