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At 6 o'clock, the tip of the hour hand was 23cm from the tip of the minute hand. At 9 o'clock, this distance was only 17cm. By how many centimeters does the minute hand's length exceed the hour hand's length?  ( no diagram provided) I tried to use Pythagorean and in the end, I didn't know what to do after I went from there. 

 Apr 30, 2019

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At 6 o'clock, the tip of the hour hand was 23cm from the tip of the minute hand. At 9 o'clock, this distance was only 17cm. By how many centimeters does the minute hand's length exceed the hour hand's length?  ( no diagram provided) I tried to use Pythagorean and in the end, I didn't know what to do after I went from there. 

 

You were on the right track with Pythagorean.  At six o'clock, the hands are in a straight line so we know the total length of the two hands is 23 cm.  One hand length we will call L so the other hand is (23 – L).

 

Draw your triangle when the hour hand is at nine and the minute hand at twelve.  One hand length is L, the other hand length is (23 – L), and the hypotenuse is 17. 

 

so, by the Pythagorean Theorem,                    L2 + (23 – L)2 = 172 

 

multiply it on out                                               L2 + (529 – 46L + L2) = 289 

 

remove parentheses and rearrange terms      2L2 – 46L + 240 = 0               (240 came from 529 minus 289)

 

this will factor to                                               (2L – 16)(L – 15) = 0 

 

set each term to zero                                       2L – 16 = 0     thus  L = 8

                                                                          L – 15 = 0     thus  L = 15

 

the minute hand is longer, so it's the one that's 15 and the hour hand is 8

 

so the minute hand length exceeds the hour hand length by 15 – 8 = 7 cm

17 cm

 May 1, 2019
 #1
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Best Answer

 

At 6 o'clock, the tip of the hour hand was 23cm from the tip of the minute hand. At 9 o'clock, this distance was only 17cm. By how many centimeters does the minute hand's length exceed the hour hand's length?  ( no diagram provided) I tried to use Pythagorean and in the end, I didn't know what to do after I went from there. 

 

You were on the right track with Pythagorean.  At six o'clock, the hands are in a straight line so we know the total length of the two hands is 23 cm.  One hand length we will call L so the other hand is (23 – L).

 

Draw your triangle when the hour hand is at nine and the minute hand at twelve.  One hand length is L, the other hand length is (23 – L), and the hypotenuse is 17. 

 

so, by the Pythagorean Theorem,                    L2 + (23 – L)2 = 172 

 

multiply it on out                                               L2 + (529 – 46L + L2) = 289 

 

remove parentheses and rearrange terms      2L2 – 46L + 240 = 0               (240 came from 529 minus 289)

 

this will factor to                                               (2L – 16)(L – 15) = 0 

 

set each term to zero                                       2L – 16 = 0     thus  L = 8

                                                                          L – 15 = 0     thus  L = 15

 

the minute hand is longer, so it's the one that's 15 and the hour hand is 8

 

so the minute hand length exceeds the hour hand length by 15 – 8 = 7 cm

17 cm

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