in the derivation of the formula for the volume of a cone, the volume of the cone is calculated to be…

in the derivation of the formula for the volume of a cone, the volume of the cone is calculated to be $\frac{pi}{4}$ times the volume of the pyramid that it fits inside. which expression represents the volume of the cone that is $\frac{pi}{4}$ times the volume of the pyramid that it fits inside? $\frac{pi}{4}(2r^{2}h)$ $\frac{pi}{4}(4r^{2}h)$ $\frac{pi}{4}(\frac{r^{2}h}{3})$ $\frac{pi}{4}(\frac{4r^{2}h}{3})$

in the derivation of the formula for the volume of a cone, the volume of the cone is calculated to be $\frac{pi}{4}$ times the volume of the pyramid that it fits inside. which expression represents the volume of the cone that is $\frac{pi}{4}$ times the volume of the pyramid that it fits inside? $\frac{pi}{4}(2r^{2}h)$ $\frac{pi}{4}(4r^{2}h)$ $\frac{pi}{4}(\frac{r^{2}h}{3})$ $\frac{pi}{4}(\frac{4r^{2}h}{3})$

Answer

Explanation:

Step1: Determine the base area of the pyramid

The pyramid's base is a square with side length (2r), so base area (= (2r)^2 = 4r^2).

Step2: Calculate the volume of the pyramid

Volume of pyramid (= \frac{1}{3} \times \text{base area} \times \text{height} = \frac{1}{3}(4r^2h)).

Step3: Relate cone volume to pyramid volume

Cone volume is (\frac{\pi}{4}) times pyramid volume: (\frac{\pi}{4} \times \frac{4r^2h}{3}).

Answer:

(\frac{\pi}{4}\left(\frac{4r^2h}{3}\right))