What is the formula for the volume of a right cone with base area [tex]B[/tex] and height [tex]h[/tex]?

A. [tex]V=-\frac{1}{3} B h[/tex]
B. [tex]V=2 B h^2[/tex]
C. [tex]V=B h[/tex]
D. [tex]V=\frac{1}{3} B h[/tex]



Answer :

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]