IDNLearn.com provides a collaborative platform for sharing and gaining knowledge. Ask anything and receive prompt, well-informed answers from our community of knowledgeable experts.
Sagot :
To determine the area [tex]\( A \)[/tex] of polygons drawn with 3 to 10 boundary dots [tex]\( (b) \)[/tex] where there are only 2 interior dots [tex]\( (i) \)[/tex], we can use Pick's Theorem. Pick's Theorem states that for a simple polygon with [tex]\( i \)[/tex] interior lattice points and [tex]\( b \)[/tex] boundary lattice points, the area [tex]\( A \)[/tex] of the polygon can be given by:
[tex]\[ A = i + \frac{b}{2} - 1 \][/tex]
Given that [tex]\( i = 2 \)[/tex], we substitute into the formula as follows:
[tex]\[ A = 2 + \frac{b}{2} - 1 = \frac{b}{2} + 1 \][/tex]
Now, let's calculate [tex]\( A \)[/tex] for each value of [tex]\( b \)[/tex] from 3 to 10.
1. For [tex]\( b = 3 \)[/tex]:
[tex]\[ A = \frac{3}{2} + 1 = 1.5 + 1 = 2.5 \][/tex]
2. For [tex]\( b = 4 \)[/tex]:
[tex]\[ A = \frac{4}{2} + 1 = 2 + 1 = 3.0 \][/tex]
3. For [tex]\( b = 5 \)[/tex]:
[tex]\[ A = \frac{5}{2} + 1 = 2.5 + 1 = 3.5 \][/tex]
4. For [tex]\( b = 6 \)[/tex]:
[tex]\[ A = \frac{6}{2} + 1 = 3 + 1 = 4.0 \][/tex]
5. For [tex]\( b = 7 \)[/tex]:
[tex]\[ A = \frac{7}{2} + 1 = 3.5 + 1 = 4.5 \][/tex]
6. For [tex]\( b = 8 \)[/tex]:
[tex]\[ A = \frac{8}{2} + 1 = 4 + 1 = 5.0 \][/tex]
7. For [tex]\( b = 9 \)[/tex]:
[tex]\[ A = \frac{9}{2} + 1 = 4.5 + 1 = 5.5 \][/tex]
8. For [tex]\( b = 10 \)[/tex]:
[tex]\[ A = \frac{10}{2} + 1 = 5 + 1 = 6.0 \][/tex]
Fill in the table with the calculated areas:
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|}
\hline
(b) & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline
(A) & 2.5 & 3.0 & 3.5 & 4.0 & 4.5 & 5.0 & 5.5 & 6.0 \\
\hline
\end{tabular}
So, the relationship between [tex]\( b \)[/tex] and [tex]\( A \)[/tex] when [tex]\( i = 2 \)[/tex] can be summarized by the formula:
[tex]\[ A = \frac{b}{2} + 1 \][/tex]
This gives us a straightforward method to determine the area for any polygon with [tex]\( i = 2 \)[/tex] interior dots by knowing only the boundary dots [tex]\( b \)[/tex].
[tex]\[ A = i + \frac{b}{2} - 1 \][/tex]
Given that [tex]\( i = 2 \)[/tex], we substitute into the formula as follows:
[tex]\[ A = 2 + \frac{b}{2} - 1 = \frac{b}{2} + 1 \][/tex]
Now, let's calculate [tex]\( A \)[/tex] for each value of [tex]\( b \)[/tex] from 3 to 10.
1. For [tex]\( b = 3 \)[/tex]:
[tex]\[ A = \frac{3}{2} + 1 = 1.5 + 1 = 2.5 \][/tex]
2. For [tex]\( b = 4 \)[/tex]:
[tex]\[ A = \frac{4}{2} + 1 = 2 + 1 = 3.0 \][/tex]
3. For [tex]\( b = 5 \)[/tex]:
[tex]\[ A = \frac{5}{2} + 1 = 2.5 + 1 = 3.5 \][/tex]
4. For [tex]\( b = 6 \)[/tex]:
[tex]\[ A = \frac{6}{2} + 1 = 3 + 1 = 4.0 \][/tex]
5. For [tex]\( b = 7 \)[/tex]:
[tex]\[ A = \frac{7}{2} + 1 = 3.5 + 1 = 4.5 \][/tex]
6. For [tex]\( b = 8 \)[/tex]:
[tex]\[ A = \frac{8}{2} + 1 = 4 + 1 = 5.0 \][/tex]
7. For [tex]\( b = 9 \)[/tex]:
[tex]\[ A = \frac{9}{2} + 1 = 4.5 + 1 = 5.5 \][/tex]
8. For [tex]\( b = 10 \)[/tex]:
[tex]\[ A = \frac{10}{2} + 1 = 5 + 1 = 6.0 \][/tex]
Fill in the table with the calculated areas:
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|}
\hline
(b) & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline
(A) & 2.5 & 3.0 & 3.5 & 4.0 & 4.5 & 5.0 & 5.5 & 6.0 \\
\hline
\end{tabular}
So, the relationship between [tex]\( b \)[/tex] and [tex]\( A \)[/tex] when [tex]\( i = 2 \)[/tex] can be summarized by the formula:
[tex]\[ A = \frac{b}{2} + 1 \][/tex]
This gives us a straightforward method to determine the area for any polygon with [tex]\( i = 2 \)[/tex] interior dots by knowing only the boundary dots [tex]\( b \)[/tex].
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! IDNLearn.com is committed to your satisfaction. Thank you for visiting, and see you next time for more helpful answers.
