Final answer:
m varies directly with the square of r. With the initial condition r=2, m=14, we find the constant of proportionality to be k=3.5. Therefore, m=504 when r=12, and r=8 when m=224.
Step-by-step explanation:
Understanding the concept of proportional relationships is vital in solving problems related to direct proportionality. The student's question implies that the quantity m varies in direct proportion to the square of r (m ∝ r²). This relationship means that the ratio of m to r² is constant. Given the initial condition that when r = 2, m = 14, we can find this constant of proportionality, k, by rearranging the formula to m = kr², which gives us k = m/r² = 14/4 = 3.5.
To solve the first part of the question, work out m when r=12: using the same formula m = kr², we plug in our known k and the given value of r and get m = 3.5 · 12² = 3.5 · 144 = 504.
The second part of the question is to work out r when m=224. We use the formula again, but this time we solve for r. Here's how it is done:
- Set the equation: 224 = 3.5r²
- Divide by k, the constant of proportionality: r² = 224/3.5
- Calculate r²: r² = 64
- Find r by taking the square root: r = √64
- Thus, r = 8