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Original Question
Error analysis: To find the distance between points (2, - 5) and (4, 10) the following steps were used. Is the distance correct...Question
Error analysis:
To find the distance between points (2, -5) and (4, 10), the following steps were used.
Is the distance correct? If the distance is not correct, where was the FIRST mistake made?
Step 1: d=(4−2)2−(10+5)2−−−−−−−−−−−−−−−−√
Step 2: d=(2)2−(15)2−−−−−−−−−−√
Step 3: d=4−225−−−−−−√
Step 4: d=−221−−−−√
Step 5: d=−14.9
(1 point)
Responses
The distance formula has addition in between the parenthesis, so step 1 should be d=(4−2)2+(10+5)2−−−−−−−−−−−−−−−−√
.
The distance formula has addition in between the parenthesis, so step 1 should be d is equal to square root of open paren 4 minus 2 close paren squared plus open paren 10 plus 5 close paren squared end root.
Step 1 should have (10−5)2
as the second parenthesis.
Step 1 should have open paren 10 minus 5 close paren squared as the second parenthesis.
In step 2, (15)2
means multiply 15 times 2 to get 30 instead of 225 for step 3.
In step 2, 15 squared means multiply 15 times 2 to get 30 instead of 225 for step 3.
There are no mistakes. The distance of -14.9 is correct.
To find the distance between points (2, -5) and (4, 10), the following steps were used.
Is the distance correct? If the distance is not correct, where was the FIRST mistake made?
Step 1: d=(4−2)2−(10+5)2−−−−−−−−−−−−−−−−√
Step 2: d=(2)2−(15)2−−−−−−−−−−√
Step 3: d=4−225−−−−−−√
Step 4: d=−221−−−−√
Step 5: d=−14.9
(1 point)
Responses
The distance formula has addition in between the parenthesis, so step 1 should be d=(4−2)2+(10+5)2−−−−−−−−−−−−−−−−√
.
The distance formula has addition in between the parenthesis, so step 1 should be d is equal to square root of open paren 4 minus 2 close paren squared plus open paren 10 plus 5 close paren squared end root.
Step 1 should have (10−5)2
as the second parenthesis.
Step 1 should have open paren 10 minus 5 close paren squared as the second parenthesis.
In step 2, (15)2
means multiply 15 times 2 to get 30 instead of 225 for step 3.
In step 2, 15 squared means multiply 15 times 2 to get 30 instead of 225 for step 3.
There are no mistakes. The distance of -14.9 is correct.
Answers
GPT-4o mini
The correct response is:
**The distance formula has addition in between the parenthesis, so step 1 should be d is equal to square root of open paren 4 minus 2 close paren squared plus open paren 10 minus (-5) close paren squared end root.**
The errors in the initial steps are as follows: In Step 1, the distance formula should correctly use addition (`+`) between the squared terms, as it calculates the square of the differences in both the x-coordinates and the y-coordinates. Moreover, the second part should be \( (10 - (-5))^2 \) instead of \( (10 + 5)^2 \). This is where the first mistake is made.
So the correct form should be:
\[ d = \sqrt{(4 - 2)^2 + (10 - (-5))^2} \]
After correcting that, you get:
\[ d = \sqrt{(2)^2 + (15)^2} \]
Which correctly simplifies to:
\[ d = \sqrt{4 + 225} = \sqrt{229} \]
This means the distance is approximately \( 15.13 \) (not \( -14.9 \)). The misunderstanding of the formula and incorrect operations led to the wrong conclusion, highlighting the importance of the correct application of the distance formula.
**The distance formula has addition in between the parenthesis, so step 1 should be d is equal to square root of open paren 4 minus 2 close paren squared plus open paren 10 minus (-5) close paren squared end root.**
The errors in the initial steps are as follows: In Step 1, the distance formula should correctly use addition (`+`) between the squared terms, as it calculates the square of the differences in both the x-coordinates and the y-coordinates. Moreover, the second part should be \( (10 - (-5))^2 \) instead of \( (10 + 5)^2 \). This is where the first mistake is made.
So the correct form should be:
\[ d = \sqrt{(4 - 2)^2 + (10 - (-5))^2} \]
After correcting that, you get:
\[ d = \sqrt{(2)^2 + (15)^2} \]
Which correctly simplifies to:
\[ d = \sqrt{4 + 225} = \sqrt{229} \]
This means the distance is approximately \( 15.13 \) (not \( -14.9 \)). The misunderstanding of the formula and incorrect operations led to the wrong conclusion, highlighting the importance of the correct application of the distance formula.