First, we establish the relationship between y, a, b, and c. We know that y varies jointly as a and b, which can be written as y = k * a * b, where k is the constant of variation. We also know that y varies inversely as the square root of c. Therefore, we can modify our equation to include this relationship as well, resulting in: y = k*a*b/sqrt(c).
Initially, we have the values a1=3, b1=2, c1=64, and y1=12. We can use these values to find our constant of variation (k).
Substituting these values into our equation gives us: 12 = k * 3 * 2 / sqrt(64). We can simplify the numerator and the denominator to get: 12 = 6k / 8. Further simplification yields: 12 = 0.75k. To isolate k, we then multiply each side of the equation by the reciprocal of 0.75 (which is 4/3), resulting in: k = 12 * 4/3 = 16.
Now that we've found k, we can use it along with the new values of a, b, and c to find the new value of y. With a2=5, b2=2, c2=25, the equation becomes: y2 = k * a2 * b2 / sqrt(c2) = 16 * 5 * 2 / sqrt(25).
This simplifies to: y2 = 16 * 10 / 5, since the square root of 25 is 5. Further simplification gives us: y2 = 160 / 5 = 32.
So, if a=5, b=2, and c=25, then y=32.