A Jet Flies At A Rate Of $1.3 \times 10^6$ Feet Per Hour. Written In Scientific Notation, Which Is The Best Estimate Of How Many Feet The Jet Will Travel In $2.8 \times 10^3$ Hours?A. $ 3 × 10 3 3 \times 10^3 3 × 1 0 3 [/tex] Feet B.

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Understanding Scientific Notation

Scientific notation is a way of expressing very large or very small numbers in a compact form. It consists of a number between 1 and 10, multiplied by a power of 10. For example, the number 456,000 can be written in scientific notation as $4.56 \times 10^5$. This notation makes it easier to perform calculations with large numbers.

The Jet's Speed

The jet flies at a rate of $1.3 \times 10^6$ feet per hour. This means that every hour, the jet covers a distance of $1.3 \times 10^6$ feet.

Calculating the Total Distance

To find the total distance the jet will travel in $2.8 \times 10^3$ hours, we need to multiply the jet's speed by the number of hours. This can be done by multiplying the two numbers in scientific notation.

Multiplying Numbers in Scientific Notation

When multiplying numbers in scientific notation, we multiply the numbers and add the exponents of the powers of 10. In this case, we have:

1.3×106×2.8×1031.3 \times 10^6 \times 2.8 \times 10^3

To multiply these numbers, we first multiply the numbers themselves:

1.3×2.8=3.641.3 \times 2.8 = 3.64

Then, we add the exponents of the powers of 10:

106×103=106+3=10910^6 \times 10^3 = 10^{6+3} = 10^9

So, the product of the two numbers in scientific notation is:

3.64×1093.64 \times 10^9

Converting to Standard Form

To convert this number to standard form, we can move the decimal point 9 places to the right, since the exponent is 9. This gives us:

3,640,000,0003,640,000,000

Conclusion

The best estimate of how many feet the jet will travel in $2.8 \times 10^3$ hours is $3.64 \times 10^9$ feet, which is equivalent to $3,640,000,000$ feet.

Answer

The correct answer is:

3.64×109\boxed{3.64 \times 10^9}

Discussion

This problem requires an understanding of scientific notation and how to multiply numbers in this notation. It also requires the ability to convert a number from scientific notation to standard form. The solution involves multiplying the jet's speed by the number of hours and then converting the result to standard form.

Additional Examples

Here are a few additional examples of multiplying numbers in scientific notation:

  • 2.5×104×3.2×102=?2.5 \times 10^4 \times 3.2 \times 10^2 = ?

  • 1.8×106×4.5×104=?1.8 \times 10^6 \times 4.5 \times 10^4 = ?

  • 3.2×103×2.1×105=?3.2 \times 10^3 \times 2.1 \times 10^5 = ?

These examples can be used to practice multiplying numbers in scientific notation and to reinforce the concept of adding exponents when multiplying numbers in this notation.

Real-World Applications

Scientific notation is used in many real-world applications, including:

  • Physics: Scientific notation is used to express large or small numbers in physics, such as the speed of light or the distance between galaxies.
  • Chemistry: Scientific notation is used to express large or small numbers in chemistry, such as the number of atoms in a molecule or the concentration of a solution.
  • Engineering: Scientific notation is used to express large or small numbers in engineering, such as the size of a building or the speed of a machine.

In all of these fields, scientific notation makes it easier to perform calculations and to express large or small numbers in a compact form.

Conclusion

In conclusion, multiplying numbers in scientific notation involves multiplying the numbers and adding the exponents of the powers of 10. This notation is used in many real-world applications, including physics, chemistry, and engineering. By understanding how to multiply numbers in scientific notation, we can perform calculations more easily and express large or small numbers in a compact form.

Understanding Scientific Notation

Scientific notation is a way of expressing very large or very small numbers in a compact form. It consists of a number between 1 and 10, multiplied by a power of 10. For example, the number 456,000 can be written in scientific notation as $4.56 \times 10^5$. This notation makes it easier to perform calculations with large numbers.

Q&A

Q: What is scientific notation?

A: Scientific notation is a way of expressing very large or very small numbers in a compact form. It consists of a number between 1 and 10, multiplied by a power of 10.

Q: How do I convert a number to scientific notation?

A: To convert a number to scientific notation, you need to move the decimal point to the left or right until you have a number between 1 and 10. Then, you multiply the number by a power of 10.

Q: What is the exponent in scientific notation?

A: The exponent in scientific notation is the power of 10 that the number is multiplied by. For example, in the number $4.56 \times 10^5$, the exponent is 5.

Q: How do I multiply numbers in scientific notation?

A: When multiplying numbers in scientific notation, you multiply the numbers and add the exponents of the powers of 10.

Q: What is the product of $1.3 \times 10^6$ and $2.8 \times 10^3$?

A: The product of $1.3 \times 10^6$ and $2.8 \times 10^3$ is $3.64 \times 10^9$.

Q: How do I convert a number from scientific notation to standard form?

A: To convert a number from scientific notation to standard form, you can move the decimal point to the left or right by the number of places indicated by the exponent.

Q: What is the standard form of $3.64 \times 10^9$?

A: The standard form of $3.64 \times 10^9$ is 3,640,000,000.

Q: What are some real-world applications of scientific notation?

A: Scientific notation is used in many real-world applications, including physics, chemistry, and engineering.

Q: Why is scientific notation useful?

A: Scientific notation is useful because it makes it easier to perform calculations with large numbers and to express large or small numbers in a compact form.

Additional Examples

Here are a few additional examples of multiplying numbers in scientific notation:

  • 2.5×104×3.2×102=?2.5 \times 10^4 \times 3.2 \times 10^2 = ?

  • 1.8×106×4.5×104=?1.8 \times 10^6 \times 4.5 \times 10^4 = ?

  • 3.2×103×2.1×105=?3.2 \times 10^3 \times 2.1 \times 10^5 = ?

These examples can be used to practice multiplying numbers in scientific notation and to reinforce the concept of adding exponents when multiplying numbers in this notation.

Conclusion

In conclusion, scientific notation is a useful tool for expressing large or small numbers in a compact form. By understanding how to multiply numbers in scientific notation, we can perform calculations more easily and express large or small numbers in a compact form.

Frequently Asked Questions

Q: What is the difference between scientific notation and standard form?

A: Scientific notation is a way of expressing numbers in a compact form, while standard form is the usual way of writing numbers.

Q: How do I convert a number from standard form to scientific notation?

A: To convert a number from standard form to scientific notation, you need to move the decimal point to the left or right until you have a number between 1 and 10, and then multiply the number by a power of 10.

Q: What is the product of $4.56 \times 10^5$ and $2.1 \times 10^3$?

A: The product of $4.56 \times 10^5$ and $2.1 \times 10^3$ is $9.616 \times 10^8$.

Q: How do I divide numbers in scientific notation?

A: When dividing numbers in scientific notation, you divide the numbers and subtract the exponents of the powers of 10.

Q: What is the quotient of $1.3 \times 10^6$ and $2.8 \times 10^3$?

A: The quotient of $1.3 \times 10^6$ and $2.8 \times 10^3$ is $4.64 \times 10^2$.

Conclusion

In conclusion, scientific notation is a useful tool for expressing large or small numbers in a compact form. By understanding how to multiply and divide numbers in scientific notation, we can perform calculations more easily and express large or small numbers in a compact form.