if square root of 8 is equal to 2.83 and square root of 11 is equal to 3.32 estimate the square root of 10 using linear interpolation.
To estimate the square root of 10 using linear interpolation, we can use the given information about the square roots of 8 and 11.
First, let's identify the range within which 10 falls. Based on the given values, we see that the square root of 8 (2.83) is smaller than the square root of 10, and the square root of 11 (3.32) is larger than the square root of 10. Therefore, 10 falls between the values of 2.83 and 3.32.
Next, we can calculate the interpolation factor, which is the ratio of the distance between 10 and 8 to the distance between 11 and 8.
Interpolation Factor = (10 - 8) / (11 - 8) = 2 / 3
Finally, we can use this interpolation factor to estimate the square root of 10.
Estimated square root of 10 ≈ square root of 8 + (Interpolation Factor * (square root of 11 - square root of 8))
Estimated square root of 10 ≈ 2.83 + (2/3) * (3.32 - 2.83)
Estimated square root of 10 ≈ 2.83 + (2/3) * 0.49
Estimated square root of 10 ≈ 2.83 + 0.33
Estimated square root of 10 ≈ 3.16
To estimate the square root of 10 using linear interpolation, you can use the given values for the square root of 8 and 11.
Linear interpolation is a method of estimating a value between two known values using a straight line. The equation for linear interpolation is:
y = y1 + ((x - x1) / (x2 - x1)) * (y2 - y1)
In this case, we want to estimate the square root of 10, which is the "x" value. The known values are the square root of 8 and 11, which are the "x1" and "x2" values, respectively. The corresponding square roots are the "y1" and "y2" values.
Let's calculate it step by step:
1. Square root of 8 = 2.83 (y1)
Square root of 11 = 3.32 (y2)
x1 = 8, x2 = 11
2. Plug the values into the linear interpolation equation:
sqrt(10) ≈ 2.83 + ((10 - 8) / (11 - 8)) * (3.32 - 2.83)
3. Simplify the equation:
sqrt(10) ≈ 2.83 + (2 / 3) * (3.32 - 2.83)
sqrt(10) ≈ 2.83 + (2 / 3) * 0.49
sqrt(10) ≈ 2.83 + (2 / 3) * 0.49
sqrt(10) ≈ 2.83 + 0.33
4. Calculate the result:
sqrt(10) ≈ 3.16
Therefore, using linear interpolation, the estimated square root of 10 is approximately 3.16.
since 10 is 2/3 of the way from 8 to 11, we want to go 2/3 of the way from √8 to √11.
that would be
2.83 + (10-8)/(11-8) * (3.32-2.83) ≈ 3.157
check: √10 = 3.162