Well, we could try adding up odd numbers, and look to see when we reach 400. But I'm hoping to find an easier way.
First of all ... I'm not sure this will help, but let's stop and notice it anyway ...
An odd number of odd numbers (like 1, 3, 5) add up to an odd number, but
an even number of odd numbers (like 1,3,5,7) add up to an even number.
So if the sum is going to be exactly 400, then there will have to be an even
number of items in the set.
Now, let's put down an even number of odd numbers to work with,and see
what we can notice about them:
1, 3, 5, 7, 9, 11, 13, 15 .
Number of items in the set . . . 8
Sum of all the items in the set . . . 64
Hmmm. That's interesting. 64 happens to be the square of 8 .
Do you think that might be all there is to it ?
Let's check it out:
Even-numbered lists of odd numbers:
1, 3 Items = 2, Sum = 4
1, 3, 5, 7 Items = 4, Sum = 16
1, 3, 5, 7, 9, 11 Items = 6, Sum = 36
1, 3, 5, 7, 9, 11, 13, 15 . . Items = 8, Sum = 64 .
Amazing ! The sum is always the square of the number of items in the set !
For a sum of 400 ... which just happens to be the square of 20,
we just need the <em><u>first 20 consecutive odd numbers</u></em>.
I slogged through it on my calculator, and it's true.
I never knew this before. It seems to be something valuable
to keep in my tool-box (and cherish always).
95141 1404 393
Answer:
- arc BC: 8.55 cm
- chord BC: 8.03 cm
Step-by-step explanation:
The length of an arc that subtends central angle α will be ...
s = rα . . . . where α is in radians
The central angle BOC is twice the measure of angle QBC, so is 70°, or 7π/18 radians. So, the length of arc BC is ...
s = (7 cm)(7π/18) ≈ 8.55 cm . . . arc BC
__
For central angle α and radius r, the chord subtending the arc is ...
c = 2r·sin(α/2)
c = 2(7 cm)sin(35°) ≈ 8.03 cm . . . . chord AB
72 divided by negative 9 is -8
