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Chapter 7: Problem 6

The radius of Earth is about 6400 kilometers. Assume that Earth is spherical. Express your answers to the questions below in scientific notation. a. Find the surface area of Earth in square meters. b. Find the volume of Earth in cubic meters. c. Find the ratio of the surface area to the volume.

Short Answer

Expert verified
(a) \(5.15 \times 10^{14} \text{ square meters}\), (b) \(1.10 \times 10^{21} \text{ cubic meters}\), (c) \(4.68 \times 10^{-7}\)

Step by step solution

01

Convert Radius to Meters

The given radius of Earth is 6400 kilometers. Convert this to meters by multiplying by 1000, since there are 1000 meters in a kilometer: \[ R = 6400 \times 1000 = 6.4 \times 10^6 \text{ meters} \]
02

Calculate Surface Area

The formula for the surface area of a sphere is \( 4\pi R^2 \). Plug in the radius in meters: \[ A = 4\pi (6.4 \times 10^6)^2 \] Calculate the square of the radius: \[ (6.4 \times 10^6)^2 = 4.096 \times 10^{13} \] So the surface area is: \[ A = 4\pi \times 4.096 \times 10^{13} \approx 5.15 \times 10^{14} \text{ square meters} \]
03

Calculate Volume

The formula for the volume of a sphere is \( \frac{4}{3}\pi R^3 \). Plug in the radius in meters: \[ V = \frac{4}{3}\pi (6.4 \times 10^6)^3 \] Calculate the cube of the radius: \[ (6.4 \times 10^6)^3 = 2.62 \times 10^{20} \] So the volume is: \[ V = \frac{4}{3}\pi \times 2.62 \times 10^{20} \approx 1.10 \times 10^{21} \text{ cubic meters} \]
04

Calculate the Ratio of Surface Area to Volume

To find the ratio of surface area to volume, divide the surface area by the volume: \[ \frac{A}{V} = \frac{5.15 \times 10^{14}}{1.10 \times 10^{21}} \approx 4.68 \times 10^{-7} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

surface area of a sphere
The surface area of a sphere is a measure of the total area that the surface of the sphere occupies.
The formula to calculate the surface area (\text{A}) of a sphere is: \[ A = 4\text{\pi}R^2 \] where \text{\pi} is approximately 3.14159 and \text{R} is the radius of the sphere.
For example, if the radius of Earth is 6.4 脳 10^6 meters, we follow these steps:
  • Square the radius: \[ (6.4 \times 10^6)^2 = 4.096 \times 10^{13} \]
  • Multiply by 4\text{\pi}: \[ 4 \pi \times 4.096 \times 10^{13} \approx 5.15 \times 10^{14} \]

    • Thus, Earth's surface area is approximately 5.15 脳 10^14 square meters. Using this method can help you solve many surface area problems effortlessly.
    volume of a sphere
    The volume of a sphere is the amount of space inside it.
    The formula to calculate the volume (\text{V}) of a sphere is: \[ V = \frac{4}{3}\text{\textbackslash\pi}R^3 \] where \text{R} is the radius of the sphere.
    Assuming the Earth's radius is 6.4 脳 10^6 meters, we can determine the volume like this:
  • Cubic the radius: \[ (6.4 \times 10^6)^3 = 2.62 \times 10^{20} \]
  • Multiply by \text{\textbackslash\frac{4}{3}\text{\textbackslash\pi}}: \[ \frac{4}{3} \times 2.62 \times 10^{20} \approx 1.10 \times 10^{21} \]
    This means the volume of Earth is about 1.10 脳 10^21 cubic meters. Mastering this formula can clarify volume calculations for any spherical object.
    • ratio calculations
    Understanding ratio calculations is important for comparing different quantities.
    The ratio of the surface area to the volume of a sphere gives us an idea about their relative sizes.
    To find this ratio, you simply divide the surface area (\text{A}) by the volume (\text{V}).
    Let鈥檚 take Earth鈥檚 surface area and volume:
  • Surface area: 5.15 脳 10^14 square meters.
  • Volume: 1.10 脳 10^21 cubic meters.

    • Divide them to find the ratio: \[\text{ratio} = \frac{5.15 \times 10^{14}}{1.10 \times 10^{21}} \approx 4.68 \times 10^{-7} \]
    Therefore, the ratio of Earth鈥檚 surface area to its volume is approximately 4.68 脳 10^{-7}. This simplified understanding of ratios can assist in solving various comparison problems easily.

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    Most popular questions from this chapter

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