Final answer:
Using the given that t varies directly as m and inversely as the square of n, a constant of variation k is determined using initial conditions and then applied to new values of m and n to find t, resulting in t = 32/3 or approximately 10.67.
Step-by-step explanation:
The question states that t varies directly as m and inversely as the square of n, and we have the condition given as t = 16 when m = 8 and n = 2. We are then asked to find t when m = 12 and n = 3. By using direct and inverse variation, we can establish a proportionality constant (k).
From the initial condition, we can write the relation as:
t = k * (m / n2)
Plugging in the initial values, we have 16 = k * (8 / 22), which simplifies to 16 = k * 2, giving us k = 8.
We can then use this proportionality constant to find the new value of t using the new values of m and n: t = k * (m / n2) = 8 * (12 / 32). This simplifies to t = 8 * (12 / 9) = 8 * (4 / 3) = 32 / 3 or approximately 10.67.