Alg 2 Word Problem

Question

The volumes of two spheres differ by 800/3 π cubic meters and the lengths of their radii differ by 5 meters. Find the length of the radius of the smaller sphere. (V=4/3πr^3)

 

Starting review for school, I have no idea how to do this.

Answer 1

 

R = radius of large sphere

r = radius of small sphere

 

Given:  

 

r = R - 5 ⇒ R - r = 5

(4/3)πR³ - (4/3)πr³ = (800/3)π ⇒ R³ - r³ = [3/(4π)] (800/3)π = 200

 

R³ - r³ = 200

 

Note that R³ - r³ is a difference of cubes.  Factor it.

 

R³ - r³ = (R - r)(R² + Rr + r²) = 5(R² + Rr + r²)

 

200 =  5(R² + Rr + r²)

 

40 = R² + Rr + r²

 

r² + R² + Rr - 40 = 0

 

r² + (r + 5)² + (r + 5)r - 40 = 0

 

r² + (r² + 10r + 25) + (r² + 5r) - 40 = 0

 

3r² + 15r - 15 = 0

 

r² + 5r - 5 = 0

 

r ≅ 0.8541 m

 

Test the solution.