To solve this problem, we can use the method of direct variation, which states that two quantities that are directly proportional can be represented by the equation y = kx, where k is the proportionality constant.
From the problem, we know that r varies directly as m² and inversely as s. So, we can represent this as the equation r = k * m² / s, where k is our proportionality constant.
Step 1: Find the constant of variation, k
We first use the initial values we know (r_initial = 15, m_initial = 9, s_initial = 9) to determine the constant k. We plug these values into our equation which gives us:
k = r_initial * s_initial / m_initial²
k = 15 * 9 / 9²
k = 1.6666666666666667
This is our constant of proportionality.
Step 2: Find the final value of r (r_final)
Now we know the value of k, we can use this alongside other given values (m_final = 45, s_final = 9) to find our final value for r.
Plugging the values into the equation r = k * m² / s, we get:
r_final = k * m_final² / s_final
r_final = 1.6666666666666667 * 45² / 9
r_final = 375
So when m is 45 and s is 9, r is 375.