use the calculator below to approximate $\\sqrt{140}$ as a decimal to the hundredths place. note that you…

use the calculator below to approximate $\\sqrt{140}$ as a decimal to the hundredths place. note that you must do the approximation without using a square root button. your answer must be within a hundredth of the actual value.

use the calculator below to approximate $\\sqrt{140}$ as a decimal to the hundredths place. note that you must do the approximation without using a square root button. your answer must be within a hundredth of the actual value.

Answer

Explanation:

Step1: Find the range

Since (11^2 = 121) and (12^2=144), (\sqrt{140}) is between (11) and (12).

Step2: Use linear approximation

Let (y = \sqrt{x}), (x_0 = 121), (\Delta x=140 - 121 = 19). The derivative (y^\prime=\frac{1}{2\sqrt{x}}), at (x = 121), (y^\prime=\frac{1}{2\times11}=\frac{1}{22}). By linear approximation formula (L(x)=y(x_0)+y^\prime(x_0)(x - x_0)), so (\sqrt{140}\approx\sqrt{121}+\frac{1}{22}(140 - 121)=11+\frac{19}{22}). Calculate (\frac{19}{22}\approx0.86). So (11 + 0.86=11.86).

Another way: Let's assume (\sqrt{140}=a + b), where (a = 11) (integer part), and ((a + b)^2=140), (a^{2}+2ab + b^{2}=140). Since (b) is small, (b^{2}\approx0). Then (121+22b=140), (22b=19), (b=\frac{19}{22}\approx0.86).

Answer:

(11.86)