The diagonal of a square equals 2–√ times the side length. Using 1.4 for the square root of 2, what is the approximate length in feet from one corner to an opposite corner of a square park that is 700 feet on a side?(1 point)
The given equation states that the diagonal of a square is equal to 2 – √2 times the side length. Let's call the side length of the square "s".
To find the length from one corner to an opposite corner (which is the diagonal), we need to multiply the side length by 2 – √2.
Given that 1.4 is the approximate value for √2, we can substitute it into the equation:
Diagonal = (2 – √2) * side length
Diagonal = (2 – 1.4) * 700
Diagonal = 0.6 * 700
Diagonal = 420
Therefore, the approximate length from one corner to an opposite corner of the square park is 420 feet.
Using a number line, what whole number and rational number to the nearest half is 58−−√ between?(1 point)
To find the whole number and rational number to the nearest half that √58 is between, we can use a number line.
First, we need to find the whole number and rational number to the left (less than) √58:
We know that 7² = 49, and 8² = 64. Since 58 is greater than 49 and less than 64, we can estimate that √58 is between 7 and 8 on the number line.
Now, let's find the rational number to the nearest half:
We can divide the range between 7 and 8 into halves. The halfway point between 7 and 8 is (7 + 8) / 2 = 7.5.
Therefore, to the nearest half, √58 is between the whole number 7 and the rational number 7.5.
Which of the following is a rational number between 5 and 5 1/4?
5 1/7
√26
5 2/5
5 1/3
To determine which of the options is a rational number between 5 and 5 1/4, we need to convert 5 and 5 1/4 to a common fraction with the same denominator.
5 is equal to 20/4 in fraction form.
5 1/4 is equal to (4*5 + 1)/4, which simplifies to 21/4.
Now, let's examine the options:
1) 5 1/7 = (7*5 + 1)/7 = 36/7, which is greater than 5 1/4.
2) √26 is an irrational number, as it involves the square root of a non-perfect square.
3) 5 2/5 = (5*5 + 2)/5 = 27/5, which is greater than 5 1/4.
4) 5 1/3 = (3*5 + 1)/3 = 16/3, which is less than both 5 and 5 1/4.
Based on this analysis, none of the options (5 1/7, √26, 5 2/5, 5 1/3) is a rational number between 5 and 5 1/4.
What is a rational number with a denominator of 7 that is between 7–√ and 8–√ . Write your answer as an improper fraction.(1 point)
To find a rational number with a denominator of 7 that is between 7 – √ and 8 – √, we can first simplify the square roots.
Let's start with 7 – √:
This cannot be simplified further.
Next, let's simplify 8 – √:
8 – √ = (√64) – √ = (√(8*8)) - √ = 8 – 2√2
Now, we have:
7 - √2 < Rational Number < 8 - 2√2
Since we are looking for a rational number with a denominator of 7, we can multiply the inequality by 7 to eliminate the square root:
7(7 - √2) < 7(Rational Number) < 7(8 - 2√2)
49 - 7√2 < 7(Rational Number) < 56 - 14√2
To find a rational number between these expressions, we need to find a rational number with a denominator of 7 between 49 - 7√2 and 56 - 14√2.
We can choose a rational number with 7 as its denominator that is between these two expressions. One possible choice is the average of these two expressions:
Rational Number = (49 - 7√2 + 56 - 14√2)/2 = (105 - 21√2)/2 = 105/2 - (21/2)√2
Therefore, the rational number with a denominator of 7 that is between 7 – √ and 8 – √ is (105/2 - (21/2)√2)/7.