Question

the volume v of a cone in terms of its radius r and height h is given by (formula partially visible).

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} ).

Explanation:

Part A

Step1: Recall Cone Volume Formula

The volume of a cone is given by \( V = \frac{1}{3}\pi r^{2}h \).

Step2: Solve for \( h \)

Multiply both sides by 3: \( 3V=\pi r^{2}h \).
Divide both sides by \( \pi r^{2} \): \( h = \frac{3V}{\pi r^{2}} \).

Step1: Identify Given Values

We know \( V = 32\pi \, \text{cm}^3 \), \( r = 4 \, \text{cm} \), and use \( h=\frac{3V}{\pi r^{2}} \).

Step2: Substitute Values

Substitute \( V = 32\pi \) and \( r = 4 \) into the formula:
\( h=\frac{3\times32\pi}{\pi\times4^{2}} \).

Step3: Simplify

Simplify \( \pi \) cancels out, \( 4^{2}=16 \), numerator: \( 96\pi \), denominator: \( 16\pi \).
\( h=\frac{96}{16}=6 \).

Answer:

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

Part B