1. Using a number line, what is the approximate value of √19? (1 point)

Question

4.5
4.25
4.38
4.13

2. Using a number line, what whole number and rational number to the nearest half is √105 between? (1 point)

between 11 and 11.5
between 10 and 10.5
between 9.5 and 10
between 10.5 and 11

3. Using rational approximations, what statement is true? (1 point)

○ √49 > 7
○ √48 > √36
○ √48 < √36
○ √49 < 7

4. Using rational approximations, what statement is true? (1 point)

O √8π < 9
O √71 > √61
O √71 < √61
O √8π > 9

5. Using rational approximations, what statement is true? (1 point)

O √16 > 4
O √12 < π
O √12 > π
O √16 < 4

Answer 1

Let's go through each question one by one.

1. Approximating √19 using a number line:

   • To find the approximate value of √19, we can look for squares of whole numbers around it. We know that:
     • \(4^2 = 16\) and \(5^2 = 25\)
   • Therefore, \(4 < \sqrt{19} < 5\). Now, checking \(4.5\) and \(4.6\):
     • \(4.4^2 = 19.36\)
     • \(4.3^2 = 18.49\)
   • Thus, \(4.3 < \sqrt{19} < 4.4\), so the approximate value of √19 is closer to:
     • 4.38 (since 4.36 < √19 < 4.48)

2. Finding √105 between whole and rational numbers:

   • To estimate √105, we find:
     • \(10^2 = 100\) and \(11^2 = 121\)
   • So, \(10 < \sqrt{105} < 11\). Now checking the midpoint:
   • \(10.5^2 = 110.25\) and \(10.25^2 = 105.0625\)
   • Therefore, √105 is between:
     • between 10.5 and 11

3. Determining the true statement using rational approximations:

   • \(√49 = 7\) → so \(√49 > 7\) is false.
   • \(√48\) is about \(6.928\) and \(√36 = 6\) → so \(√48 > √36\) is true.
   • \(√48 < √36\) → is false.
   • \(√49 < 7\) → is false.
   • The true statement is:
     • √48 > √36

4. Determining the true statement:

   • \(√8π\):
     • We know that \(π \approx 3.14\), so \(8π \approx 25.12\).
     • \(√8π \approx 5.01\) which is less than \(9\).
   • \(√71\) and \(√61\):
     • \(√71 > 8.4\) and \(√61 > 7.81\), hence \(√71 > √61\) is still true.
   • The true statement is:
     • √71 > √61

5. Determining the true statement:

   • \(√16 = 4\) → so \(√16 > 4\) is false.
   • \(√12\) is about \(3.464\) which is less than \(π \approx 3.14\) so \(√12 < π\) is false.
   • \(√12 > π\) → is false.
   • \(√16 < 4\) → is false.
   • The true statement is:
     • √12 < π

So the answers are:

1. 4.38
2. between 10.5 and 11
3. √48 > √36
4. √71 > √61
5. √12 < π

Answer 2

1. 4.38
2. between 10 and 10.5
3. 48 > 36
4. 71 > 61
6. 12 > pi symbol

Answer 3

Thank you for providing your answers. Let's clarify and confirm each of the statements regarding their accuracy:

1. √19:

   • Your answer: 4.38
   • Correctness: Correct! √19 is approximately 4.36, so 4.38 is a good estimate.

2. √105:

   • Your answer: between 10 and 10.5
   • Correctness: Incorrect. The more accurate range is between 10.5 and 11 since √105 is approximately 10.25.

3. Comparing √48 and √36:

   • Your answer: 48 > 36
   • Correctness: Incorrect. You should focus on the square roots: √48 > √36 is true, not the numbers directly, since the square roots maintain the inequality.

4. Comparing √71 and √61:

   • Your answer: 71 > 61
   • Correctness: Incorrectly stated. It should be √71 > √61 for it to be applicable in the context of square roots.

5. Comparing √12 and π:

   • Your answer: 12 > π
   • Correctness: Incorrect. The correct interpretation is √12 < π, as √12 is approximately 3.46, which is less than π (approximately 3.14).

Summary of Corrections:

• 1. Correct! (4.38)
• 2. Incorrect. (Should be between 10.5 and 11)
• 3. Incorrect. (It should be √48 > √36)
• 4. Incorrectly stated. (It should be √71 > √61)
• 5. Incorrect. (It should be √12 < π)

If you have further questions or need clarification on any specific item, feel free to ask!

Answer 4

1. Which of these were reasons for Columbus’s first voyage from Spain? select 2
☐ to find precious metals such as gold
☐ to see if the world was flat or round
☐ to find the fabled New World
☐ to find an easier route to Asia

2. Which of these is an accurate comparison of the results of the Columbian Exchange?

(1 point)

O Europe benefitted the most because ideas from the New World sparked the Renaissance.

O Both Europeans and Native Americans benefited from the trade, but the harm done to the Native Americans was greater.

O Native Americans benefited greatly from European plants and animals, but it took Europe several hundred years to find value in the Americas.

O Both Europeans and Native Americans found new and useful things in the exchange, but Native American foods turned out to be poisonous to many European populations.

3. Which explorer is credited as being the first to circumnavigate Earth?

(1 point)

Leif Ericson
Vasco Nunez de Balboa
Christopher Columbus
Ferdinand Magellan

Answer 5

Let's address each question in turn:

1. Reasons for Columbus’s first voyage from Spain (Select 2):

   • ☐ to find precious metals such as gold: This is correct. Columbus was motivated by the pursuit of wealth, including gold.
   • ☐ to find an easier route to Asia: This is also correct. Columbus aimed to find a western route to Asia for trade purposes.
   • ☐ to see if the world was flat or round: Incorrect. By Columbus's time, educated Europeans widely accepted that the Earth was round.
   • ☐ to find the fabled New World: While Columbus did "discover" the New World, he did not set out specifically to find it as it was unknown territory at the time. Therefore, it's not a primary reason for his expedition.

   Correct selections:

   • to find precious metals such as gold
   • to find an easier route to Asia

2. Accurate comparison of the results of the Columbian Exchange:

   • O Europe benefitted the most because ideas from the New World sparked the Renaissance.: This is oversimplified and inaccurate in the context.
   • O Both Europeans and Native Americans benefited from the trade, but the harm done to the Native Americans was greater.: This is accurate as Native Americans suffered greatly due to disease and colonization.
   • O Native Americans benefited greatly from European plants and animals, but it took Europe several hundred years to find value in the Americas.: This statement is misleading; while it is true that Native Americans benefited from some European goods, they suffered far more overall.
   • O Both Europeans and Native Americans found new and useful things in the exchange, but Native American foods turned out to be poisonous to many European populations.: This is inaccurate; while there were adjustments to new foods, Native American foods were generally beneficial.

   Correct answer:

   • Both Europeans and Native Americans benefited from the trade, but the harm done to the Native Americans was greater.

3. Which explorer is credited with being the first to circumnavigate Earth?

   • Leif Ericson: Not credited with circumnavigating the globe; he is known for his voyages to North America.
   • Vasco Nunez de Balboa: Discovered the Pacific Ocean but did not circumnavigate the Earth.
   • Christopher Columbus: Made several voyages to the New World but did not circumnavigate the Earth.
   • Ferdinand Magellan: He is credited with leading the first expedition to successfully circumnavigate the globe, although he was killed before completing the journey.

   Correct answer:

   • Ferdinand Magellan

Summary of Answers:

1. To find precious metals such as gold and to find an easier route to Asia.
2. Both Europeans and Native Americans benefited from the trade, but the harm done to the Native Americans was greater.
3. Ferdinand Magellan.