Question

A cone has a volume V, a radius r, and a height of 12 cm. a.) A cone has the same height and of the radius of the original cone. Write an expression for it's volume. b.) A cone has the same height and 3 times the radius of the original cone. Write an expression for its volume.

Answer 1

expressionGiven from the original,

Volume=V; radius= r and height = 12cm

Please note that the volume V, of a cone, is calculated with the formula

$\begin{gathered} The\text{Volume of a Cone= }\frac{1}{3}\times Circular\text{ Base Area}\times\text{Height} \\ =\frac{1}{3}\times\Pi r^2\times h \end{gathered}$

For question (a)

height is the same as the original but the radius is

$\text{new radius,R= }\frac{1}{3}of\text{ r=}\frac{1}{3}\times r=\frac{r}{3}$

The expression for the new volume will give us

$\begin{gathered} \text{Note, V}_{cone}=\frac{1}{3}\times\Pi R^2\times H \\ H=h=12;\text{ R=}\frac{r}{3} \\ The\text{ New Volume=}\frac{1}{3}\times\Pi\times(\frac{r}{3})^2\times12 \\ =\frac{1}{3}\times3.14\times\frac{r^2}{9}\times12 \\ =\frac{37.68r^2h}{27} \\ =1.396r^2cm^3 \end{gathered}$

(a) Therefore, The expression for the new volume is 1.396r²cm³

(b) At this time, the height has the height as the original cone but 3 times the radius of the original cone.

Note that the new height still remains as h, while the new radius will be

$\begin{gathered} \text{New height= h} \\ Then\text{ew radius= 3}\times r=3r \end{gathered}$

Substituting the new height and the new radius into the formula for the volume of the cone will give

$\begin{gathered} V_{cone}=\frac{1}{3}\times\Pi\times r^2\times h \\ \text{the new radius =3r and the new height is still h} \\ V_{cone}=\frac{1}{3}\times\Pi\times(3r)^2\times12 \\ =\frac{1}{3}\times3.14\times9r^2\times12 \\ =\frac{3.14\times9r^2\times12}{3} \\ =113.04r^2cm^3 \end{gathered}$

(b) Therefore, the expression for the new volume is 113.04r²cm³