To determine the correct formula for the volume of a right cone with base area [tex]\( B \)[/tex] and height [tex]\( h \)[/tex], let's consider each of the given choices and evaluate them.
### Step-by-step Evaluation:
1. Choice A: [tex]\( V = -\frac{1}{3} B h \)[/tex]
This formula suggests that the volume [tex]\( V \)[/tex] is negative, which is not physically meaningful in the context of volume. Therefore, choice A is incorrect.
2. Choice B: [tex]\( V = 2 B h^2 \)[/tex]
This formula suggests that the volume [tex]\( V \)[/tex] is proportional to the square of the height [tex]\( h \)[/tex] and also includes a factor of 2. The actual volume of a right cone does not have such a dependency. Hence, choice B is also incorrect.
3. Choice C: [tex]\( V = B h \)[/tex]
This formula suggests that the volume [tex]\( V \)[/tex] is directly proportional to both the base area [tex]\( B \)[/tex] and the height [tex]\( h \)[/tex]. While this might seem plausible at first, it does not account for the fact that a cone is only a fraction of what would be a cylinder with the same base area and height. Volume of a right cone is actually one-third of the volume of such a cylinder. Therefore, choice C is incorrect.
4. Choice D: [tex]\( V = \frac{1}{3} B h \)[/tex]
This formula correctly accounts for the fact that the volume of a cone is one-third of the volume of a cylinder with the same base area [tex]\( B \)[/tex] and height [tex]\( h \)[/tex]. Thus, this matches the well-known formula for the volume of a right cone.
### Conclusion:
The correct formula for the volume of a right cone with base area [tex]\( B \)[/tex] and height [tex]\( h \)[/tex] is:
[tex]\[ V = \frac{1}{3} B h \][/tex]
Therefore, the correct answer is:
D. [tex]\( V = \frac{1}{3} B h \)[/tex]
