Certainly! Let's solve this step-by-step.
We know from the problem statement that:
- Each man has a mass ([tex]\( m_1 \)[/tex] and [tex]\( m_2 \)[/tex]) of [tex]\( 90 \, kg \)[/tex].
- The gravitational force ([tex]\( F \)[/tex]) between them is [tex]\( 8.64 \times 10^{-8} \, N \)[/tex].
- The gravitational constant ([tex]\( G \)[/tex]) is [tex]\( 6.67 \times 10^{-11} \, N \cdot (m/kg)^2 \)[/tex].
We want to find the distance ([tex]\( r \)[/tex]) between the two men. The formula for gravitational force is:
[tex]\[ F = G \cdot \frac{m_1 \cdot m_2}{r^2} \][/tex]
Solving for [tex]\(\ r \)[/tex]:
[tex]\[ r^2 = G \cdot \frac{m_1 \cdot m_2}{F} \][/tex]
First, let's compute the value of [tex]\( r^2 \)[/tex]:
[tex]\[ r^2 = \frac{G \cdot m_1 \cdot m_2}{F} \][/tex]
[tex]\[ r^2 = \frac{6.67 \times 10^{-11} \, N \cdot (m/kg)^2 \cdot 90 \, kg \cdot 90 \, kg}{8.64 \times 10^{-8} \, N} \][/tex]
Given that:
[tex]\[ G = 6.67 \times 10^{-11} \, N \cdot (m/kg)^2 \][/tex]
[tex]\[ m_1 = 90 \, kg \][/tex]
[tex]\[ m_2 = 90 \, kg \][/tex]
[tex]\[ F = 8.64 \times 10^{-8} \, N \][/tex]
Plugging in these values:
[tex]\[ r^2 = \frac{(6.67 \times 10^{-11}) \cdot 90 \cdot 90}{8.64 \times 10^{-8}} \][/tex]
[tex]\[ r^2 \approx 6.253125 \][/tex]
Next, we need to find the square root of [tex]\(\ r^2 \)[/tex] to get [tex]\( r \)[/tex]:
[tex]\[ r = \sqrt{6.253125} \][/tex]
[tex]\[ r \approx 2.5006 \, m\][/tex]
Therefore, the distance between the two men is approximately [tex]\(2.5\, m\)[/tex].
So, the correct answer is:
C. [tex]\(2.5\, m\)[/tex]
