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 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 \(r\).

Key Concepts

Direct Variation

Direct variation describes a relationship where one variable is proportional to another. In this problem, r varies directly as the square of m, meaning r increases as m² increases, expressed as r ∝ m².

Inverse Variation

Inverse variation means one variable changes in the opposite way to another. Here, r varies inversely as s, so as s increases, r decreases proportionally, represented as r ∝ 1/s.

Formulating and Solving Variation Equations

To solve, combine direct and inverse variations into an equation: r = k * (m²) / s, where k is a constant. Use given values to find k, then substitute new values of m and s to find r.