Using a number line, what is the approximate value of 19−−√?(1 point)
Responses
4.25
4.5
4.13
4.38
To find the approximate value of √19 using a number line, we can first approximate the square root of 19 as a whole number. Since the square of 4 is 16 and the square of 5 is 25, we know that √19 is between 4 and 5 on the number line.
Next, we can estimate the decimal part of √19 by dividing the number line segment between 4 and 5 into smaller parts. Since there are 10 equally spaced divisions between 4 and 5, each division represents a value of 0.1.
To estimate the decimal part of √19, we can count the number of divisions it would take to reach √19 on the number line. It would take approximately 3.8 divisions to reach √19, which can be written as 3.8 * 0.1 = 0.38.
Adding the estimated whole number value of 4 to the estimated decimal value of 0.38, we get an approximate value of √19 as 4.38.
Therefore, the answer is 4.38.
Using a number line, what whole number and rational number to the nearest half is 105−−−√ between?(1 point)
Responses
between 10 and 10.5
between 9.5 and 10
between 10.5 and 11
between 11 and 11.5
To determine the whole number and rational number to the nearest half that √105 is between using a number line, we can first find the closest perfect squares to 105. The square of 10 is 100, and the square of 11 is 121.
Since √105 is between the perfect squares of 100 and 121, we know that it falls between the whole numbers of 10 and 11 on the number line.
Next, we can estimate the decimal part of √105 by dividing the number line segment between 10 and 11 into halves.
To estimate the decimal part of √105, we can count the number of divisions it would take to reach √105 on the number line. It would take approximately 0.5 divisions to reach √105.
Adding the estimated whole number value of 10 to the estimated decimal value of 0.5, we get an approximate value of √105 as 10.5.
Therefore, √105 is between 10 and 11 on the number line, and to the nearest half, it is between 10.5 and 11.
Thus, the answer is between 10.5 and 11.
Using rational approximations, what statement is true?(1 point)
Responses
49−−√<7
48−−√<36−−√
49−−√>7
48−−√>36−−√
To determine which statement is true using rational approximations, we can first find the square roots of 49 and 48 using a calculator or by simplifying the radical if possible.
The square root of 49 is 7 because 7 * 7 = 49.
The square root of 48 is approximately 6.9282 when rounded to four decimal places.
Comparing the statements:
1. 49−−√<7: This statement is false because the square root of 49 is equal to 7, not less than 7.
2. 48−−√<36−−√: This statement is true because the square root of 48 (approximately 6.9282) is less than the square root of 36 (6).
3. 49−−√>7: This statement is false because the square root of 49 is equal to 7, not greater than 7.
4. 48−−√>36−−√: This statement is true because the square root of 48 (approximately 6.9282) is greater than the square root of 36 (6).
Therefore, the statement "48−−√<36−−√" is true.
Using rational approximations, what statement is true?(1 point)
Responses
71−−√>61−−√
start root 71 end root greater than start root 61 end root
81−−√<9
start root 81 end root less than 9
81−−√>9
start root 81 end root greater than 9
71 −−−√<61−−√
To determine which statement is true using rational approximations, we can first find the square roots of 71 and 61 using a calculator or by simplifying the radicals if possible.
The square root of 71 is approximately 8.4261 when rounded to four decimal places.
The square root of 61 is approximately 7.8102 when rounded to four decimal places.
Comparing the statements:
1. 71−−√>61−−√: This statement is true because the square root of 71 (approximately 8.4261) is greater than the square root of 61 (approximately 7.8102).
2. 81−−√<9: This statement is false because the square root of 81 is equal to 9, not less than 9.
3. 81−−√>9: This statement is false because the square root of 81 is equal to 9, not greater than 9.
4. 71−−√<61−−√: This statement is false because the square root of 71 (approximately 8.4261) is greater than the square root of 61 (approximately 7.8102), not less than.
Therefore, the statement "71−−√>61−−√" is true.
Using rational approximations, what statement is true?(1 point)
Responses
16−−√>4
start root 16 end root greater than 4
12 −−−√<π
start root 12 end root less than pi
12 −−−√ >π
start root 12 end root greater than pi
16 −−−√<4
To determine which statement is true using rational approximations, we can compare the square root of 16 with the value 4 and compare the square root of 12 with the value of π (pi).
The square root of 16 is exactly 4, which is the same as the value given in the statement "16−−√>4."
The square root of 12 is approximately 3.4641 when rounded to four decimal places.
Comparing the statements:
1. 16−−√>4: This statement is false because the square root of 16 is exactly 4, not greater than 4.
2. 12−−√<π: This statement is true because the square root of 12 (approximately 3.4641) is less than the value of π (approximately 3.1416).
3. 12−−√>π: This statement is false because the square root of 12 (approximately 3.4641) is less than the value of π (approximately 3.1416), not greater than.
4. 16−−√<4: This statement is false because the square root of 16 is exactly 4, not less than 4.
Therefore, the statement "12−−√<π" is true.