Final answer:
When M is inversely proportional to r squared and M is 160 when r is 4, and we want to find r when M is 540, we first find the constant of proportionality with the initial values, then use that constant to find the new value of r. The result is approximately r = 2.18.
Step-by-step explanation:
The question you've asked is based on the concept of inverse variation in mathematics. This concept states that when one variable increases, the other variable decreases at a rate that keeps the product of the two variables constant. In this case, the problem states that M is inversely proportional to the square of r (M ∙ r² = k, where k is the constant of proportionality). When M = 160, r is given as 4, so we can use this to find the constant k:
k = M ∙ r² = 160 ∙ 4² = 160 ∙ 16 = 2560
Now, when M is 540, we need to find the new value of r. We use the equation with our constant:
540 ∙ r² = 2560
r² = 2560 / 540
r² ≈ 4.74
r ≈ √4.74
r ≈ 2.18
Therefore, when m is 540, the value of r is approximately 2.18.