正比與反比

2014-05-11 10:04 pm
1.為一同心圓圓形靶,靶上的各區域的分數為12、4、1分,若各區域內得分數與其區域面積成反比,則由內而外,三區域面積比為何?
2.承上題,這三個同心圓的半徑比為何?

回答 (6)

2014-05-11 10:54 pm
✔ 最佳答案
............................

2014-05-11 14:58:07 補充:
1.a1:a2:a3

=pi*r1^2:pi(r2^2-r1^2):pi(r3^2-r2^2)

=1:4:12.......ans

2014-05-11 15:02:10 補充:
2.承上題,這三個同心圓的半徑比為何?

a1/1=a2/4=a3/12=k

a1=k=pi*r1^2 => r1^2=k/pi

a2=4k

=pi*(r^2-r1^2)

r2^2=a2/pi+r1^2

=4k/pi+k/pi

=5k/pi

a3=pi(r3^2-r2^2)

r3^2=a3/pi+r2^2

=12k/pi+5k/pi

=17k/pi

令k=pi

r1=1(cm)

r2=√5(cm)=2.236(cm)

r3=√17(cm)=4.1231(cm).......ans

2014-05-11 18:58:04 補充:
(1) 補充:

r1,r2,r3=小,中,大圓半徑

a1=小圓面積=pi*r1^2

a2=中圓面積=pi*(r2^2-r1^2)

a3=大圓面積=pi*(r3^2-r2^2)

=> a1:a2:a3=r1^2:(r2^2-r1^2):(r3^2-r2^2)

=反比例

=1:4:12

2014-05-11 19:01:42 補充:
(2) 補充:

連比例可以改為正比例:

a1/1=a2/4=a3/12=k

=> a1=k, a2=4k, a3=12k

a1=k=pi*r1^2 => r1^2=k/pi

a2=4k=pi*(r^2-r1^2)

=> r2^2=a2/pi+r1^2

=4k/pi+k/pi

=5k/pi

2014-05-11 19:05:46 補充:
a3=12k=pi*(r3^2-r2^2)

=> r3^2=a3/pi+r2^2

=12k/pi+5k/pi

=17k/pi

r1^2:r2^2:r3^2=k:5k:17k=1:5:17

=> r1:r2:r3=1:√5:√17

2014-05-12 04:17:14 補充:
1.a1:a2:a3

=pi*r1^2:pi(r2^2-r1^2):pi(r3^2-r2^2)

=1:4:12.......ans

2014-05-12 04:18:12 補充:
上面張貼錯誤

(1) 修正:

a1:a2:a3=反比

=(1/12):(1/4):1

=(12/12):(12/4):12.....同時*12

=1:3:12

2014-05-12 04:18:40 補充:
(2) 修正

正比改為連比:

a1/1=a2/3=a3/12=k^2 => a1=k^2, a2=3k^2, a3=12k^2


k^2=r1^2 => r1=k


3k^2=r2^2-r1^2 => r2^2=3k^2+r1^2=4k^2 => r2=2k


12k^2=r3^2-r2^2 => r3^2=12k^2+4k^2=16k^2 => r3=4k


=> r1:r2:r3=1:2:4
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