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

In step 1, 82.5−−−−√
is between 8 and 10 because 82.5−−−−√∼80
and 8×10=80
.

Answer 1

Hal did not make an error. However, let's clarify the details:

1. Step 1: Hal correctly stated that since \( 9^2 = 81 \) and \( 10^2 = 100 \), \( \sqrt{82.5} \) is indeed between 9 and 10.

2. Step 2: Hal calculated the squares of \( 9.0 \), \( 9.1 \), and \( 9.2 \) correctly. He noted that \( 81 < 82.5 < 82.81 \), which is accurate.

3. Step 3: Hal proceeded to square values closer to \( 9.1 \), \( 9.08 \) and \( 9.09 \), which correctly bracket \( 82.5 \) with \( 82.44 < 82.5 < 82.62 \).

4. Step 4: Since \( 82.5 \) is closer to \( 82.44 \) than to \( 82.62 \), Hal's estimation of \( 9.09 \) as the best approximation is indeed valid.

Given this analysis, Hal did not make any errors in his estimation process. Therefore, the correct response is:

Hal did not make an error.