Volume of cone is given by formula
-------------------------------------------(1)
We are already given r =2, so we need to find h and then we can find volume
So missing meausre is height here ------------------------------------------------------------------------------
To find h we will use lateral area =
given
Formula for lateral area of cone is ![\pi r\sqrt{r^{2}+ h^{2}}](https://tex.z-dn.net/?f=%20%5Cpi%20r%5Csqrt%7Br%5E%7B2%7D%2B%20h%5E%7B2%7D%7D%20%20%20%20)
so we have![12 = \pi r\sqrt{r^{2}+ h^{2}}](https://tex.z-dn.net/?f=%2012%20%3D%20%5Cpi%20r%5Csqrt%7Br%5E%7B2%7D%2B%20h%5E%7B2%7D%7D%20%20%20%20)
Now plug r as 2 in this equation
![12 \pi = \pi (2)\sqrt{2^{2}+ h^{2}}](https://tex.z-dn.net/?f=%2012%20%5Cpi%20%3D%20%5Cpi%20%282%29%5Csqrt%7B2%5E%7B2%7D%2B%20h%5E%7B2%7D%7D%20%20%20%20)
Now solve for h as shown and get h by itself
![6 = \sqrt{4+h^{2}}](https://tex.z-dn.net/?f=%206%20%20%3D%20%5Csqrt%7B4%2Bh%5E%7B2%7D%7D%20%20%20%20%20%20)
![6^{2} = \sqrt{4+h^{2}} ^{2}](https://tex.z-dn.net/?f=%206%5E%7B2%7D%20%20%20%3D%20%5Csqrt%7B4%2Bh%5E%7B2%7D%7D%20%20%5E%7B2%7D%20%20)
![36} = 4+h^{2}](https://tex.z-dn.net/?f=%2036%7D%20%20%20%3D%204%2Bh%5E%7B2%7D%20%20)
![36 - 4 = 4+h^{2} -4](https://tex.z-dn.net/?f=%2036%20-%204%20%20%20%20%3D%204%2Bh%5E%7B2%7D%20-4%20)
![32 = h^{2}](https://tex.z-dn.net/?f=%2032%20%20%3D%20h%5E%7B2%7D%20%20)
![\sqrt{32} = \sqrt{h^{2}}](https://tex.z-dn.net/?f=%20%5Csqrt%7B32%7D%20%3D%20%5Csqrt%7Bh%5E%7B2%7D%7D%20%20%20)
![\sqrt{16 \times 2} = h](https://tex.z-dn.net/?f=%20%5Csqrt%7B16%20%5Ctimes%202%7D%20%3D%20h%20%20)
![\sqrt{16} \times \sqrt{2} = h](https://tex.z-dn.net/?f=%20%5Csqrt%7B16%7D%20%5Ctimes%20%5Csqrt%7B2%7D%20%3D%20h%20%20)
----------------------------------------------------(2)
Now plug 2 in r place and
in h place in volume formula given in (1)
![V = \frac{\pi r^{2}h}{3}](https://tex.z-dn.net/?f=%20V%20%3D%20%5Cfrac%7B%5Cpi%20r%5E%7B2%7Dh%7D%7B3%7D%20%20%20%20%20)
![V = \frac{\pi (2)^{2}4\sqrt{2}}{3} [/tex][tex] V = \frac{\pi (4)4\sqrt{2}}{3}](https://tex.z-dn.net/?f=%20V%20%3D%20%5Cfrac%7B%5Cpi%20%282%29%5E%7B2%7D4%5Csqrt%7B2%7D%7D%7B3%7D%20%20%20%20%20%5B%2Ftex%3C%2Fp%3E%3Cp%3E%5D%5Btex%5D%20V%20%3D%20%5Cfrac%7B%5Cpi%20%284%294%5Csqrt%7B2%7D%7D%7B3%7D%20%20%20%20%20)
![V = \frac{16 \pi \sqrt{2}}{3}](https://tex.z-dn.net/?f=%20V%20%3D%20%5Cfrac%7B16%20%5Cpi%20%5Csqrt%7B2%7D%7D%7B3%7D%20%20%20%20%20)
so choice (1) is right answer
![(1) \frac{16 \sqrt{2}\pi}{3}](https://tex.z-dn.net/?f=%20%281%29%20%20%5Cfrac%7B16%20%5Csqrt%7B2%7D%5Cpi%7D%7B3%7D%20%20%20%20%20)
correct answer --------------------------------------------------------------------------------------------------
![(2) 16 \sqrt{2} \pi](https://tex.z-dn.net/?f=%20%282%29%20%2016%20%5Csqrt%7B2%7D%20%5Cpi%20%20)
incorrect answer as correct volume answer we got as
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![(3) 4 \sqrt{38} \pi](https://tex.z-dn.net/?f=%20%283%29%20%204%20%5Csqrt%7B38%7D%20%5Cpi%20%20)
incorrect answer as correct volume answer we got as
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