al 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 82.62 than it is to 82.44, 9.09 is the best approximation for 82.5−−−−√
.

In which step did Hal make an error?



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(1 point)
Responses

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.

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

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

In step 4, he made an error in determining which value is closer to 82.5.
In step 4, he made an error in determining which value is closer to 82.5.

Answer 1

Hal made an error in step 4 when he determined which value is closer to 82.5.

Here's a breakdown:

• In step 2, he correctly identified that \(9.0^2 = 81.00\) and \(9.1^2 = 82.81\), showing that \(81.00 < 82.5 < 82.81\).
• In step 3, he squares the hundredths: \(9.08^2 = 82.44\) and \(9.09^2 = 82.62\), which was also correct, showing that \(82.44 < 82.5 < 82.62\).
• However, in step 4, he incorrectly concluded that 82.62 is closer to 82.5. The correct determination is that 82.5 is exactly halfway between 82.44 and 82.62, so the average would be considered; however, since it is equidistant, it does not favor 9.09 over 9.08. Thus, he should have stated that either value could be right but ultimately should round differently based on the context.

So, the correct response is:

In step 4, he made an error in determining which value is closer to 82.5.