Final answer:
The value of M is 20 when r = 2, and the value of r is 6 when M = 540, using the direct proportionality relationship between M and r³.
Step-by-step explanation:
Since M is directly proportional to r³, we have M = kr³ where k is the constant of proportionality. Given that when r = 4, M = 160, we can find k by substituting these values into the equation, giving us k = 160 / 4³ = 160 / 64 = 2.5.
To find the value of M when r = 2, we substitute r = 2 into the equation M = 2.5×r³, which gives M = 2.5×2³ = 2.5×8 = 20.
For finding the value of r when M = 540, we use M = 2.5r³ and solve for r, which gives r³ = 540 / 2.5 = 216, and taking the cube root of both sides gives us r = ∛216 = 6. Therefore, the value of r when M = 540 is 6.