Question

part a
choose the formula for the height of a cone in terms of its volume and radius.
○ a ( h = \frac{r^2}{3vpi} )
○ b ( h = \frac{vpi r^2}{3} )
○ c ( h = \frac{pi r^2}{3v} )
● d ( h = \frac{3v}{pi r^2} )

part b
find the height ( h ), in centimeters, of a cone with volume ( v = 32pi , \text{cm}^3 ) and radius ( r = 4 , \text{cm} )
(square)

Explanation:

Part A

The formula for the volume \( V \) of a cone is \( V=\frac{1}{3}\pi r^{2}h \). To solve for \( h \), we first multiply both sides by 3: \( 3V = \pi r^{2}h \). Then divide both sides by \( \pi r^{2} \), giving \( h=\frac{3V}{\pi r^{2}} \), which matches option D.

Step 1: Recall the formula for height of a cone

We use the formula \( h=\frac{3V}{\pi r^{2}} \) from Part A.

Step 2: Substitute the given values

Given \( V = 32\pi\space cm^{3} \) and \( r = 4\space cm \), substitute into the formula:
\( h=\frac{3\times(32\pi)}{\pi\times(4)^{2}} \)

Step 3: Simplify the expression

First, simplify the numerator and denominator. The \( \pi \) in the numerator and denominator cancels out. \( 3\times32 = 96 \) and \( 4^{2}=16 \). So we have \( h=\frac{96}{16} \)

Step 4: Calculate the result

\( \frac{96}{16}=6 \)

Answer:

D. \( h = \frac{3V}{\pi r^{2}} \)

Part B