Given Kepler's third law, which states that [tex]\( P^2 = k \cdot A^3 \)[/tex], we need to find the value of the constant [tex]\( k \)[/tex] for Mars. We have the following given values:
- Orbital period of Mars ([tex]\( P \)[/tex]): 687 days
- Semimajor axis of Mars ([tex]\( A \)[/tex]): 1.52 AU
According to Kepler's third law:
[tex]\[ P^2 = k \cdot A^3 \][/tex]
First, we will square the orbital period ([tex]\( P \)[/tex]):
[tex]\[ 687^2 \][/tex]
Then we will cube the semimajor axis ([tex]\( A \)[/tex]):
[tex]\[ 1.52^3 \][/tex]
To isolate [tex]\( k \)[/tex], we rearrange the equation to:
[tex]\[ k = \frac{P^2}{A^3} \][/tex]
Substituting the known values into the equation:
[tex]\[ k = \frac{687^2}{1.52^3} \][/tex]
Calculating [tex]\( 687^2 \)[/tex]:
[tex]\[ 687^2 = 471969 \][/tex]
Calculating [tex]\( 1.52^3 \)[/tex]:
[tex]\[ 1.52^3 = 3.511328 \][/tex]
Finally, we divide the squared orbital period by the cubed semimajor axis to find [tex]\( k \)[/tex]:
[tex]\[ k = \frac{471969}{3.511328} \approx 134394.87580186615 \][/tex]
From the given options, the closest value to this result is:
C. [tex]\( 1.34 \times 10^5 \)[/tex]
So, the correct answer is:
C. [tex]\( 1.34 \times 10^5 \)[/tex]
