I really don't understand how to do this...

Question

The volume of several planets in cubic kilometers can be modeled by v(d)=1/6πd^3, where d is the diameter of the planet in kilometers. The mass of each planet in kilograms in terms of diameter d can be modeled by M(d)=(3.96 x 10^12)d^3 - (6.50 x 10^17)d^2 + (2.56 x 10^22)d - 5.56 x 10^25. 

 

A. The density of a planet in kilograms per cubic kilometer can be found by dividing the planet'a mass by its volume. Use polynomial division to find a model model for the density of a planet in terms of it's diameter.

 

B. Use the model to estimate the density of Jupiter, with diameter d= 142,984 km. 

Answer 1

As the problem states, density is mass divided by volume. You're given a crazy formula for mass:

M(d)=(3.96×10¹²)d³ - (6.50×10¹⁷)d² + (2.56×10²²)d - 5.56×10²⁵

 

And the formula for volume:

V(d)=(1/6)πd³

 

They want you to divide M(d) ÷ V(d). Since V(d) is a monomial (just one term), you can divide each term individually.

density = [ (3.96×10¹²)d³ - (6.50×10¹⁷)d² + (2.56×10²²)d - 5.56×10²⁵ ] / [ (1/6)πd³ ]

 = (3.96×10¹²)d³ / [ (1/6)πd³ ] - (6.50×10¹⁷)d² / [ (1/6)πd³ ] + (2.56×10²²)d / [ (1/6)πd³ ] - 5.56×10²⁵ / [ (1/6)πd³ ]

 

Divide the numbers (including π), and reduce the d's appropriately.

 

For part B, you then plug in the value of d given.

 

I hope this helps. If you have more questions, please comment and ask.