Question

A well-known feature of our solar system is the asteroid belt between Mars and Jupiter. One theory about the asteroid belt is that it's made of primordial material that was prevented from forming another planet by the gravitational pull of Jupiter when the solar system was formed. One of the largest asteroids is 951 Gaspra. Its distance from the Sun is 207.16 million miles. Use your linear regression equation to interpolate the length of its sidereal year. Please take the natural log of Distance before you plug it in, and that your first result will be the natural log of Year. Show your work.

Planet Distance from Sun(millions of miles) Years(as a fraction of Earth years) In(Dist) In(Year)

Mercury 36.19 0.2410 3.5889 -1.4229

Venus 67.63 0.6156 4.21240 -0.4851

Earth 93.5 1.007 4.5380 0.0007

Mars 142.46 1.8821 4.9591 0.6324

Jupiter 486.46 11.8704 6.1871 2.4741

Saturn 893.38 29.4580 6.7950 3.3830

Uranus 1,794.37 84.0100 7.4924 4.4209

Neptune 2,815.19 164.7800 7.9428 5.1046

Pluto 3,695.95 248.5400 8.2150 5.5156

Also part B...Calculate the length of the year for 951 Gaspra

The linear regression equation is y = 4.53 + 0.665ln(year)

Answer 1

The linear regression equation is y = 4.53 + 0.665ln(year)

=> In(Dist) = 4.53 + 0.665ln(year)

For 951 Gaspra, Dist = 207.16 million miles

In(207.16) = 4.53 + 0.665ln(year)

=> 0.665ln(year) = 5.3335 - 4.53

=> ln(year) = 0.8035 / 0.665

=> ln(year) =  1.2083

=> year = exp(1.2083) =  3.348 Years(as a fraction of Earth years)