Please answer the 3 questions on the right as thoroughly as possible. This is your chance to connect the concepts you've learned in this unit to real-world situations. Be sure to review the rubric for details on how your responses will be evaluated.

Question

Open response questions
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The diameter of Earth is approximately 1.27 x 10⁷ meters.

The tallest building in the world, the Burj Khalifa in Dubai, has a height of about 8.28 x 10² meters.

Question at position 1
1

5 points



Question at position 1
Explain whether the number 1.27 x 10⁷ is rational or irrational, and justify your reasoning.

Explain whether the number 1.27 x 10⁷ is rational or irrational, and justify your reasoning.

p

Question at position 2
2

5 points



Question at position 2
Express the diameter of Earth in standard form (not scientific notation).

Answer 1

Question 1

1. Explain whether the number 1.27 x 10⁷ is rational or irrational, and justify your reasoning.

The number \( 1.27 \times 10^7 \) is a rational number. A rational number is defined as any number that can be expressed as a fraction \( \frac{p}{q} \), where \( p \) and \( q \) are integers, and \( q \) is not equal to zero. To justify this, we can convert \( 1.27 \times 10^7 \) into a fraction.

The number \( 1.27 \) can be written as \( \frac{127}{100} \). Consequently, when we multiply this by \( 10^7 \) (which is equal to \( 10,000,000 \) or \( \frac{10,000,000}{1} \)), we can express this product in the following manner:

\[ 1.27 \times 10^7 = \frac{127}{100} \times \frac{10,000,000}{1} = \frac{127 \times 10,000,000}{100} \]

Here, we compute the numerator:

\[ 127 \times 10,000,000 = 1,270,000,000 \]

We can then write the entire expression as:

\[ \frac{1,270,000,000}{100} \]

Since both \( 1,270,000,000 \) and \( 100 \) are integers, and \( 100 \) is not zero, this confirms that \( 1.27 \times 10^7 \) can indeed be expressed as a fraction of two integers. Therefore, it is a rational number.

Question 2

2. Express the diameter of Earth in standard form (not scientific notation).

To express the diameter of Earth, which is given as \( 1.27 \times 10^7 \) meters, in standard decimal form (rather than scientific notation), we can convert it from scientific notation by moving the decimal point to the right. The number \( 1.27 \times 10^7 \) means we take \( 1.27 \) and move the decimal point seven places to the right.

Starting with \( 1.27 \):

1. Move the decimal one place to the right: \( 12.7 \)
2. Move the decimal two places to the right: \( 127 \)
3. Move the decimal three places to the right: \( 1270 \)
4. Move the decimal four places to the right: \( 12700 \)
5. Move the decimal five places to the right: \( 127000 \)
6. Move the decimal six places to the right: \( 1270000 \)
7. Move the decimal seven places to the right: \( 12700000 \)

Putting this all together, \( 1.27 \times 10^7 \) in standard decimal form is \( 12,700,000 \) meters.

Therefore, the diameter of Earth expressed in standard form is 12,700,000 meters.

Answer 2

Compare the diameter of Earth and the height of the Burj Khalifa using exponents. Explain which one is larger and by how many powers of 10.

Answer 3

To compare the diameter of Earth and the height of the Burj Khalifa using exponents, we first express both measurements in scientific notation:

• Diameter of Earth: \( 1.27 \times 10^7 \) meters
• Height of the Burj Khalifa: \( 8.28 \times 10^2 \) meters

Step 1: Identify Powers of 10

We will look at the powers of 10 associated with each measurement:

• The diameter of Earth is associated with \( 10^7 \).
• The height of the Burj Khalifa is associated with \( 10^2 \).

Step 2: Compare the Exponents

To determine which one is larger, we compare the exponents:

• \( 10^7 \) (Earth's diameter) is greater than \( 10^2 \) (Burj Khalifa's height).

Step 3: Determine the Difference in Powers of 10

To find out by how many powers of 10 the diameter of Earth is larger than the height of the Burj Khalifa, we subtract the smaller exponent from the larger exponent:

\[ 7 - 2 = 5 \]

Conclusion

The diameter of Earth is larger than the height of the Burj Khalifa by 5 powers of 10. In other words, \( 1.27 \times 10^7 \) meters (the diameter of Earth) is \( 100,000 \) times bigger than \( 8.28 \times 10^2 \) meters (the height of the Burj Khalifa), indicating a significant difference in scale between these two measurements.