To determine the value of the constant [tex]\( k \)[/tex] using Kepler's third law, we start with the known values for the semimajor axis [tex]\( A \)[/tex] and the orbital period [tex]\( P \)[/tex] of Mars. According to Kepler's third law, the relationship between the orbital period ([tex]\( P \)[/tex]) and the semimajor axis ([tex]\( A \)[/tex]) is given by the equation:
[tex]\[ P^2 = k \cdot A^3 \][/tex]
Given:
[tex]\[ A = 1.52 \, \text{AU} \][/tex]
[tex]\[ P = 687 \, \text{days} \][/tex]
We need to solve for [tex]\( k \)[/tex]. Rearrange the equation to solve for [tex]\( k \)[/tex]:
[tex]\[ k = \frac{P^2}{A^3} \][/tex]
First, calculate [tex]\( P^2 \)[/tex]:
[tex]\[ P^2 = 687^2 = 471969 \][/tex]
Next, calculate [tex]\( A^3 \)[/tex]:
[tex]\[ A^3 = (1.52)^3 \approx 3.518208 \][/tex]
Now, we can find [tex]\( k \)[/tex]:
[tex]\[ k = \frac{471969}{3.518208} \approx 134394.875 \][/tex]
After calculating, we need to match this value with the provided options:
A. [tex]\( 4.52 \times 10^2 \)[/tex]
B. [tex]\( 7.44 \times 10^{-6} \)[/tex]
C. [tex]\( 1.34 \times 10^5 \)[/tex]
D. [tex]\( 2.21 \times 10^{-3} \)[/tex]
We compare our calculated value [tex]\( 134394.875 \)[/tex] with the given options. The option that closely matches [tex]\( 134394.875 \)[/tex] is:
C. [tex]\( 1.34 \times 10^5 \)[/tex]
Therefore, the value of the constant [tex]\( k \)[/tex] for Mars is:
[tex]\[ \boxed{1.34 \times 10^5} \][/tex]
Hence, the correct answer is option C.
