Question
A graphic wants to know the number of regions that are formed when circle overlap in a particular way. can she find a rule that describes how the number of regions increases when another circle is added to the design? How many regions would a design with 6 circles create?
Number of regions
1 3 7 13 21
Type an integer or a demical
A as the number of circles increases the number of regions increases by a successive multiples of __ each time
B As the number of circles increases the number of regions increased by _ each time
C as the number of circles increases the number of regions increases by a factor of __ each time
Number of regions
1 3 7 13 21
Type an integer or a demical
A as the number of circles increases the number of regions increases by a successive multiples of __ each time
B As the number of circles increases the number of regions increased by _ each time
C as the number of circles increases the number of regions increases by a factor of __ each time
Answers
Answered by
GPT-4o mini
To analyze the number of regions formed by overlapping circles, we can look at the pattern in the given sequence of regions:
- 1 circle: 1 region
- 2 circles: 3 regions
- 3 circles: 7 regions
- 4 circles: 13 regions
- 5 circles: 21 regions
Let's find the differences between the number of regions for successive circles:
- 3 - 1 = 2
- 7 - 3 = 4
- 13 - 7 = 6
- 21 - 13 = 8
Now, we can see the first differences: 2, 4, 6, 8.
Next, let's find the second differences:
- 4 - 2 = 2
- 6 - 4 = 2
- 8 - 6 = 2
The second differences are constant (equal to 2), which indicates that the number of regions is described by a quadratic function.
An appropriate formula for the number of regions \( R(n) \) formed by \( n \) circles is:
\[
R(n) = n^2 - n + 2
\]
Now, we can use this formula to find the number of regions created by 6 circles:
\[
R(6) = 6^2 - 6 + 2 = 36 - 6 + 2 = 32
\]
### Answering the questions:
A. As the number of circles increases, the number of regions increases by a successive multiple of **2** each time (the second differences are constant).
B. As the number of circles increases, the number of regions increases by **2, 4, 6, 8, etc.** each time (the first differences).
C. As the number of circles increases, the number of regions increases by a factor of **not applicable**; the increase is not multiplicative but rather additive in a quadratic manner.
Thus, the number of regions created with 6 circles is **32**.
- 1 circle: 1 region
- 2 circles: 3 regions
- 3 circles: 7 regions
- 4 circles: 13 regions
- 5 circles: 21 regions
Let's find the differences between the number of regions for successive circles:
- 3 - 1 = 2
- 7 - 3 = 4
- 13 - 7 = 6
- 21 - 13 = 8
Now, we can see the first differences: 2, 4, 6, 8.
Next, let's find the second differences:
- 4 - 2 = 2
- 6 - 4 = 2
- 8 - 6 = 2
The second differences are constant (equal to 2), which indicates that the number of regions is described by a quadratic function.
An appropriate formula for the number of regions \( R(n) \) formed by \( n \) circles is:
\[
R(n) = n^2 - n + 2
\]
Now, we can use this formula to find the number of regions created by 6 circles:
\[
R(6) = 6^2 - 6 + 2 = 36 - 6 + 2 = 32
\]
### Answering the questions:
A. As the number of circles increases, the number of regions increases by a successive multiple of **2** each time (the second differences are constant).
B. As the number of circles increases, the number of regions increases by **2, 4, 6, 8, etc.** each time (the first differences).
C. As the number of circles increases, the number of regions increases by a factor of **not applicable**; the increase is not multiplicative but rather additive in a quadratic manner.
Thus, the number of regions created with 6 circles is **32**.
There are no human answers yet. A form for humans to post answers is coming very soon!