The solid shape is a cone and a half sphere. So the equation would be (cone volume) + (1/2 sphere volume):
[tex] cone = \frac{1}{3} \pi {r}^{2} h \\ sphere = \frac{4}{3} \pi {r}^{3} \\ object = ( \frac{1}{3} \pi {r}^{2} h) + \frac{1}{2} ( \frac{4}{3} \pi {r}^{3} )[/tex]
If the volume of the cone is 270pi and the height of the cone is 10, then:
[tex]270\pi = \frac{1}{3} \pi {r}^{2} (10) \\ 27 = \frac{1}{3} {r}^{2} \\ 80 = {r}^{2} \\ r = 9[/tex]
The radius of the object is 9 cm. Therefore, we can plug into the formula for the total volume:
[tex]object = 270\pi + \frac{1}{2} ( \frac{4}{3} \pi {(9)}^{3} ) \\ = 270\pi + \frac{2}{3} (729\pi) \\ = 270\pi \: + 243\pi \\ = 513\pi[/tex]
The total volume is 513 pi cm^3
