Final answer:
To find the volume of the cone, we can use the formula V = (1/3)πr²h, where r is the radius and h is the height. Given the volume of cone c as 12 cubic millimeters, the volume of the new cone with one-third the radius and triple the height is also 12 cubic millimeters.
Step-by-step explanation:
To find the volume of the cone, we can use the formula V = (1/3)πr²h, where r is the radius and h is the height. Let's denote the radius and height of cone c as r_c and h_c, respectively. Given that the volume of cone c is 12 cubic millimeters, we have (1/3)πr_c²h_c = 12.
Now, we need to find the volume of the cone with one-third the radius and triple the height. Denoting the radius and height of this cone as r_new and h_new, we have (1/3)πr_new²h_new = (1/3)π(r_c/3)²(3h_c) = (1/3)π(r_c²/9)(3h_c) = (1/3)πr_c²h_c = 12. So, the volume of the new cone is also 12 cubic millimeters.