Let's analyze the relationship between [tex]\( b \)[/tex] and [tex]\( c \)[/tex] for each given option by working through a sample value. Suppose [tex]\( b = 4 \)[/tex] for our demonstration:
1. Option 1: [tex]\( c \)[/tex] is 1.5 times that of [tex]\( b \)[/tex].
[tex]\[
c = 1.5 \times b = 1.5 \times 4 = 6.0
\][/tex]
So, if [tex]\( c \)[/tex] is 6.0, then [tex]\( c \)[/tex] is 1.5 times [tex]\( b \)[/tex].
2. Option 2: [tex]\( c \)[/tex] is double [tex]\( b \)[/tex].
[tex]\[
c = 2 \times b = 2 \times 4 = 8
\][/tex]
So, if [tex]\( c \)[/tex] is 8, then [tex]\( c \)[/tex] is double [tex]\( b \)[/tex].
3. Option 3: [tex]\( c \)[/tex] is the square of half of [tex]\( b \)[/tex].
[tex]\[
c = \left(\frac{b}{2}\right)^2 = \left(\frac{4}{2}\right)^2 = 2^2 = 4.0
\][/tex]
So, if [tex]\( c \)[/tex] is 4.0, then [tex]\( c \)[/tex] is the square of half of [tex]\( b \)[/tex].
4. Option 4: [tex]\( c \)[/tex] is the square of [tex]\( b \)[/tex].
[tex]\[
c = b^2 = 4^2 = 16
\][/tex]
So, if [tex]\( c \)[/tex] is 16, then [tex]\( c \)[/tex] is the square of [tex]\( b \)[/tex].
From our calculations, we have the following results for [tex]\( c \)[/tex] when [tex]\( b = 4 \)[/tex]:
- For [tex]\( c \)[/tex] being 1.5 times [tex]\( b \)[/tex], [tex]\( c = 6.0 \)[/tex].
- For [tex]\( c \)[/tex] being double [tex]\( b \)[/tex], [tex]\( c = 8 \)[/tex].
- For [tex]\( c \)[/tex] being the square of half of [tex]\( b \)[/tex], [tex]\( c = 4.0 \)[/tex].
- For [tex]\( c \)[/tex] being the square of [tex]\( b \)[/tex], [tex]\( c = 16 \)[/tex].
The results match the answer given:
[tex]\[
(6.0, 8, 4.0, 16)
\][/tex]
Thus, we have verified each of the options relative to [tex]\( b = 4 \)[/tex] and their corresponding results for [tex]\( c \)[/tex]. This approach can be used to understand the various relationships between [tex]\( b \)[/tex] and [tex]\( c \)[/tex] for different formulations.
