Question

geometry question

Answer 1

A cone of height '\(h\)' is cut by plane into two parts from the height '\(\dfrac{h}{3}\)' from the base ,and a new small cone and a frustum are formed.
Find the ratio of volume of new cone and the frustum.

Answer 2

first of all

write the equation for the volume of a cone

I will help further if oyu do that

Answer 3

\(\large \color{black}{\begin{align}V=\dfrac{\pi r^2h}{3}\hspace{.33em}\\~\\
\end{align}}\)

Answer 4

OK

So now look at he tip of the cone - the cut off bit

what is its height (compared to the original)?

Answer 5

\(\large \color{black}{\begin{align}h_1=\dfrac{2h}{3}\hspace{.33em}\\~\\
\end{align}}\)

Answer 6

ok

now - you can see that the radius of the base of the cut off bit is proportional to the height

so what is the radius of the base of the cut off bit

Answer 7

\(\large \color{black}{\begin{align}r_1=\dfrac{2r}{3}\hspace{.33em}\\~\\
\end{align}}\)

Answer 8

good

so now ypou have a small cone with h = 2H/3 and r = 2R/3

so what is the volume of this small cone?

Answer 9

\(\large \color{black}{\begin{align}V_1=\dfrac{\pi}{3}\left(\dfrac{2r}{3}\right)^2\left(\dfrac{2h}{3}\right)\hspace{.33em}\\~\\
\end{align}}\)

Answer 10

so this is 8/27 of the original

so what is the volume of the frustrum? (it's what's left when you take away 8/27)

Answer 11

\(\large \color{black}{\begin{align}V_{frustrum}=\dfrac{\pi}{3}r^2h-\dfrac{\pi}{3}\left(\dfrac{2r}{3}\right)^2\left(\dfrac{2h}{3}\right)\hspace{.33em}\\~\\
\end{align}}\)

Answer 12

YES - but it is easier than that to work out

if the tip is pir^2h/3 *(2/3)^3

that is 8/27th of th eoriginal - so how much is left in the frustrum?

Answer 13

\(\large \color{black}{\begin{align}V_{frustrum}=\dfrac{19}{27}\hspace{.33em}\\~\\
\end{align}}\) ?

Answer 14

that is not quite correct

we are talking about ratios

8/27 of the original is in the tip
19/27 of the original is in the frustrum

so what is the ratio of tip to frustrum?

Answer 15

\(\large \color{black}{\begin{align}\dfrac{V_{new cone}}{V_{frustrum}}=\dfrac{8}{19}\hspace{.33em}\\~\\
\end{align}}\)

Answer 16

correct - well done