Question

Which is the best estimate of StartRoot 0. 96 EndRoot to the nearest hundredth? 0. 96 0. 97 0. 98 1. 0.

Answer 1

The best estimate for $\sqrt{0.96}$ is 0.98. The correct option is C. 0.98.

Given expression is $\sqrt{0.96}$.

We have to estimate the nearest hundredth value of $\sqrt{0.96}$.

Now we have to first calculate the square root of 0.96 after that we can estimate the nearest hundredth value of it.

For calculating the square root of 0.96 we have various method, but here we use the 'Babylonian Method' to get the square root of any positive number.

We must set an error for the final result.

Say, smaller than 0.01. In other words we will try to find the square root value with at least 1 correct decimal places.

Babylonian Method

Step 1:

Divide the number (0.96) by 2 to get the first guess for the square root . First guess will be $\frac{0.96}{2} =0.48$

Step 2:

Divide 0.96 by the previous result, $\frac{0.96}{0.48} = 2.$

Average the value of step 2) with that of step 1:

$\frac{0+0.48}{2} = 1.24$ (new guess).

Now,

$error= new\ guess-previous \ value$

$error= 1.24-0.48\\error=0.76$

0.76 > 0.01.

As error > accuracy, we repeat this step again.

Step 3:

Divide 0.96 by the previous result.

$\frac{0.96}{1.24} =0.7741935484$

Average the above value with that of step 2:

$\frac{(0.7741935484 + 1.24)}{2} = 1.0070967742\ (new \ guess).$

$Error = new \ guess - previous \ value \\Error=1.24-1.0070967742\\Error= 0.2329032258$

0.2329032258 > 0.01.

As error > accuracy, we repeat this step again.

Step 4:

Divide 0.96 by the previous result.

$\frac{0.96}{1.0070967742} = 0.9532351057.$

Average the above value with that of step 3:

$\frac{(0.9532351057 + 1.0070967742)}{2} = 0.98016594\ (new\ guess).$ $Error = 1.0070967742 - 0.98016594 \\Error= 0.0269308342.$

0.0269308342 > 0.01.

As error > accuracy, we repeat the step again.

Step 5:

Divide 0.96 by the previous result.

$\frac{0.96}{0.98016594} = 0.9794259939.$

Average the above value with that of step 4:

$)\frac{(0.9794259939 + 0.98016594)}{2} = 0.979795967 (\ new \ guess).$ $Error = 0.98016594 - 0.979795967\\Error = 0.000369973.$

0.000369973 < 0.01.

As error < accuracy, we stop the iterations and use 0.979795967 as the square root of 0.96.

So, we can say that the square root of 0.96 is 0.979 with an error smaller than 0.01 (in fact the error is 0.000369973). this means that the first 3 decimal places are correct.

Since the value of the right side from the hundredth place is greater than 4 so for rounding off we add 1 to the hundredth place.

So the nearest hundredth value of $\sqrt{0.96}$ is 0.98.

Hence the correct option is C. 0.98.

For more details on square root follow the link:

brainly.com/question/1527773