One of Kepler’s three laws of planetary motion states that the square of the period, P, of a body orbiting the sun is proportional to the cube of its average distance, d, from the sun. The earth has a period of 365 days and its distance from the sun is approximately 93,000,000 miles. (a) Find P as a function of d. (b) The planet Jupiter has an average distance from the sun of 483,000,000 miles. How long in earth days is a Jupiter year?

In 1619, the German astronomer Johannes Kepler showed that the square of the period of a planet (how long it takes to go around the Sun) is directly proportional to the cube of of the semimajor axis of its eliptical orbit (how far it is from the Sun) with p² = D³ If the distance of the comet ATLAS from the Sun is 26.9158 AU, what is its period around the sun? Round to 2 decimal places. P = years

During Ricardo noticed an ink stain with a diameter of about 3 cm on the front of his teacher’s shirt. He then noticed that there was a pen in the pocket and that the ink blot was continuing to spread across the front of the teacher’s shirt. He estimated that the diameter of the ink blot seemed to be growing by about 0.5 cm every 2 minutes. a) Write an equation for the RADIUS, r, of the ink stain, as a function of time, t. Assume that t = 0 represents the time that Ricardo first noticed the ink stain. b) Write an equation showing the AREA, A, of the ink stain as a function of the time t. (Hint: It might help to first write A as a function of r. ) c) Draw a graph of the function A(t) for 10 minutes. d) At what time is the area of the ink stain about 25 square centimeter? Show how you answer this question. (i.e. ”I found it using desmos” is not sufficient)

Solve the following problems. A windmill blade with a diameter of 70 feet completes 6 rotations per minute. Calculate the linear speed (in 1. feet per minute) of the windmill blade.