Question

c. find the capacity of the hemisphere section of the piñata. round your answer to the nearest whole number. hint: v_sphere = \frac{4}{3}\pi r^{3} and you are working with half of a sphere
d. find the capacity of the cone section of the piñata. round your answer to the nearest whole number. hint: the total capacity of the piñata minus the capacity of the hemisphere will be the capacity of the cone section.
e. find the height of the cone using the capacity that you calculated in part (d). round your answer to one decimal place. hint: v_cone = \frac{1}{3}\pi r^{2}h

Explanation:

Step1: Find volume of hemisphere

The volume formula for a sphere is $V_{sphere}=\frac{4}{3}\pi r^{3}$. For a hemisphere, $V_{hemisphere}=\frac{1}{2}\times\frac{4}{3}\pi r^{3}=\frac{2}{3}\pi r^{3}$. But since the radius $r$ is not given in the problem - statement, assume we are given $r$. Let's say $r = 3$ (for demonstration purposes). Then $V_{hemisphere}=\frac{2}{3}\pi(3)^{3}=\frac{2}{3}\pi\times27 = 18\pi\approx 57$ (rounded to the nearest whole - number).

Step2: Find volume of cone

We know that $V_{total}=V_{hemisphere}+V_{cone}$. So $V_{cone}=V_{total}-V_{hemisphere}$. Again, since $V_{total}$ is not given, assume $V_{total} = 100$ (for demonstration). Then $V_{cone}=100 - 57=43$ (rounded to the nearest whole - number).

Step3: Find height of cone

The volume formula for a cone is $V_{cone}=\frac{1}{3}\pi r^{2}h$. We know $V_{cone}$ from step 2 and assume $r = 3$. Rearranging the formula for $h$ gives $h=\frac{3V_{cone}}{\pi r^{2}}$. Substituting $V_{cone}=43$ and $r = 3$: $h=\frac{3\times43}{\pi\times(3)^{2}}=\frac{129}{9\pi}=\frac{129}{9\times3.14}\approx4.6$ (rounded to one decimal place).

Answer:

c. 57 (assuming $r = 3$)
d. 43 (assuming $V_{total}=100$ and $r = 3$)
e. 4.6 (assuming $V_{cone}=43$ and $r = 3$)