Final answer:
The new radius of the star is approximately 18.79 km when the period decreases from 2.9 s to 2.4 s.
Step-by-step explanation:
The period of a rotating object is inversely proportional to the square of its radius. If the period of a 15.0 km radius star is initially 2.9 s and it decreases to 2.4 s, we can calculate the new radius using the period-radius relationship.
Let's assume that the initial period P_initial is equal to 2.9 s and the initial radius R_initial is equal to 15.0 km. Similarly, the final period P_final is equal to 2.4 s.
Using the formula P = kR^2, where k is a constant, we can set up the following proportion:
P_initial = kR_initial^2
P_final = kR_final^2
Dividing the two equations:
P_initial / P_final = (kR_initial^2) / (kR_final^2)
Simplifying further:
(P_initial / P_final) = (R_initial^2) / (R_final^2)
Substituting the given values:
(2.9 s / 2.4 s) = (15.0 km)^2 / (R_final^2)
Now, we can solve for R_final:
R_final^2 = (15.0 km)^2 * (2.4 s / 2.9 s)
R_final^2 = (15.0 km)^2 * 0.827586207
R_final^2 = 353.4482759 km^2
Taking the square root of both sides:
R_final = sqrt(353.4482759) km
R_final = 18.79 km
Therefore, the new radius of the star is approximately 18.79 km.