Question

Pluto orbits the sun in about 250 years. from kepler’s third law, calculate its semimajor axis. show your work. the accuracy you can easily get by estimating the roots by hand rather than with a calculator is sufficient.

Answer 1

To find the semimajor axis of Pluto using Kepler's third law, square its orbital period (250 years) to get 62500, and then estimate the cube root of that number, which is roughly 40. Thus, Pluto's semimajor axis is about 40 AU.

Step-by-step explanation:

To calculate the semimajor axis of Pluto using Kepler's third law, we apply the relationship that the square of the orbital period (P) is proportional to the cube of the semimajor axis (a). Given that Pluto's orbital period is 250 years, we calculate the square of this period: P² = 250² = 62500. Now, we need to find the cube root of this value to determine the semimajor axis.

To estimate the cube root of 62500 by hand, we can look for a number that when cubed, is close to 62500. Through trial and error or estimation, we can deduce that the number is approximately 40, since 40 × 40 × 40 = 64000, which is close to 62500. Therefore, Pluto's semimajor axis is approximately 40 astronomical units (AU).