Solve each problem. Suppose r varies directly as the square of m, and inversely as s. If r=12 when m=6 and s=4, find r when m=6 and s=20.

Verified step by step guidance

1. Identify the variation relationship given: r varies directly as the square of m and inversely as s. This can be written as the equation \(r = k \frac{m^2}{s}\), where \(k\) is the constant of proportionality.
2. Use the given values \(r=12\), \(m=6\), and \(s=4\) to find the constant \(k\). Substitute these values into the equation: \$12 = k \frac{6^2}{4}$.
3. Simplify the expression inside the fraction: \$6^2 = 36\(, so the equation becomes \)12 = k \frac{36}{4}$.
4. Solve for \(k\) by multiplying both sides by 4 and then dividing by 36: \(k = \frac{12 \times 4}{36}\).
5. With \(k\) found, substitute \(m=6\) and \(s=20\) into the original formula \(r = k \frac{m^2}{s}\) to find the new value of \(r\). Write the expression as \(r = k \frac{6^2}{20}\) and simplify accordingly.

Key Concepts

Direct Variation

Direct variation describes a relationship where one variable changes proportionally with another. If r varies directly as the square of m, it means r = k * m² for some constant k. This concept helps establish how r depends on m in the problem.

Inverse Variation

Inverse variation means one variable changes inversely as another, expressed as r = k / s for some constant k. In this problem, r varies inversely as s, indicating that as s increases, r decreases proportionally, which is key to forming the equation.

Finding the Constant of Variation

To solve variation problems, you first find the constant k by substituting known values into the variation equation. Here, using r=12, m=6, and s=4 allows calculation of k, which is then used to find r for new values of m and s.