Hal used the following procedure to find an estimate for 82.5−−−−√

Question

.

Step 1: Since 92=81
and 102=100
and 81<82.5<100
, then 82.5−−−−√
is between 9 and 10.

Step 2: Since 82.5 is closer to 81, square the tenths closer to 9

9.02=81.00


9.12=82.81


9.22=84.64


Step 3: Since 81.00<82.5<82.81
, square the hundredths closer to 9.1

9.082=82.44


9.092=82.62


Step 4: Since 82.5 is closer to 82.44 than it is to 82.62, 9.09 is the best approximation for 82.5−−−−√
.

Did Hal make an error? If so, in which step?

(1 point)
Responses

Hal did not make an error.
Hal did not make an error.

In step 1, 82.5−−−−√
is between 8 and 10 because 82.5−−−−√∼80
and 8×10=80
.
In step 1, square root of 82 point 5 is between 8 and 10 because square root of 82 point 5 approximates 80 and 8 times 10 is equal to 80.

In step 4, he made an error in selecting the correct decimal approximation.
In step 4, he made an error in selecting the correct decimal approximation.

In step 2, he made a calculation error when squaring.

Answer 1

Hal did not make an error in his procedure. However, the best option that indicates where there might be a misunderstanding in his approximation process is:

In step 4, he made an error in selecting the correct decimal approximation.

While Hal's steps generally lead him to a reasonable approximation of \(\sqrt{82.5}\), the final selection of 9.09 may not be the most precise choice based on the calculations provided.

Specifically, based on the squared values, 9.08 gives 82.44 and 9.09 gives 82.62, so since 82.5 is closer to 82.44, 9.08 would have actually been a better approximation than 9.09—leading to the conclusion that there is a potential misstep in selecting the closest value in the final step.