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 82.62 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.

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.

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

Answer 1

Hal made an error in the interpretation of the proximity of 82.5 to the squared values in step 4.

In step 3, he correctly calculated that:

• \( 9.08^2 = 82.44 \)
• \( 9.09^2 = 82.62 \)

However, when determining which of these values is closer to 82.5, he stated that 82.5 is closer to 82.62 than to 82.44. This is incorrect. The difference is computed as follows:

• \( 82.5 - 82.44 = 0.06 \)
• \( 82.62 - 82.5 = 0.12 \)

Since 0.06 is less than 0.12, 82.5 is actually closer to 82.44. Therefore, his best approximation should have been 9.08 instead of 9.09.

Thus, the correct answer is: In step 4, he made an error in determining which value is closer to 82.5.