To solve the problem of determining the value that should be substituted for [tex]\( c \)[/tex] in the equation for the centripetal force, [tex]\( X \)[/tex], we need to use the provided equation:
[tex]\[
X = \frac{b^2 a}{c}
\][/tex]
Given:
- Velocity, [tex]\( b = 12 \)[/tex] meters/second
- Radius, [tex]\( a = 2 \)[/tex] meters
Let's substitute these values into the equation:
[tex]\[
X = \frac{(12)^2 \cdot 2}{c}
\][/tex]
[tex]\[
X = \frac{144 \cdot 2}{c}
\][/tex]
[tex]\[
X = \frac{288}{c}
\][/tex]
We need to determine the correct value of [tex]\( c \)[/tex] from the given options that will make the equation hold. The options for [tex]\( c \)[/tex] are:
A. 2
B. 5
C. 6
D. 12
Let's consider each option for [tex]\( c \)[/tex]:
1. If [tex]\( c = 2 \)[/tex]:
[tex]\[
X = \frac{288}{2} = 144
\][/tex]
2. If [tex]\( c = 5 \)[/tex]:
[tex]\[
X = \frac{288}{5} = 57.6
\][/tex]
3. If [tex]\( c = 6 \)[/tex]:
[tex]\[
X = \frac{288}{6} = 48
\][/tex]
4. If [tex]\( c = 12 \)[/tex]:
[tex]\[
X = \frac{288}{12} = 24
\][/tex]
To verify our calculations, we check if the calculated value matches the scenario where the velocity and radius are consistent with the centripetal force derived by the equation provided. Based on our initial substitution:
[tex]\[
X = \frac{12^2 \cdot 2}{c}
\][/tex]
We conclude that the value of [tex]\( c \)[/tex] that makes the equation true when solved for [tex]\( X \)[/tex] with [tex]\( b = 12 \)[/tex] and [tex]\( a = 2 \)[/tex] is:
[tex]\[
\boxed{2}
\][/tex]
