calculusgeometryvolume
I need to find the volume of a metal cone that is hollow with a thickness of 2cm. The radius of the big cone is 8cm and the height is 12. The thickness determents the radius, and height of the little cone inside. How do i find the radius and height of the little cone so i can correctly find the volume of the metal?
(The cone has a base and is not like a ice-cream cone)
$$V = (\frac{1}{3}\pi R^2 H) – (\frac{1}{3}\pi r^2 h)$$
Image showing cone inside a cone:
Hint Consider the bottom empty cone with so-far unknown height $h'$ and radius $r'$. We know that is has half the volume of the whole cone, i.e. $$\frac{21 \pi}{2}= \frac{\pi (r')^2 h'}{3}.$$
Also, the bottom cone has the same angle as the whole cone, which means that $\frac{r}{h}=\frac{r'}{h'}$.
Can you use this to solve for $h'$?
First, let us draw out a diagram with using the information given. Notice that I've drawn the ice cream cone as if looking at it from the side (which in this case won't change the measurements we're interested in). Notice that I've used the 28cm twice since it is both pertinent to the height of the ice cream and also to one leg of a triangle. Notice that in the diagram, we have formed a triangle with a missing side length which I'll label as $x$. Notice that if we can find $x$, then we can find the height of the entire ice cream which will be $x+28$.
Notice that in the diagram, we have formed a triangle with legs $x$ and $28$ and hypotenuse $100$. By the Pythagorean theorem, we have that $$x^2+28^2 = 100^2.$$
Solving for $x$, $$x = \sqrt{100^2 - 28^2} = 96.$$
And thus the height is $x+28 = 96+28 = 124$.
Best Answer
A vertical cross section of the solid gives two similar triangles, one inside the other. The base of the larger triangle is $16$ and its height is $12$; the base of the smaller triangle is $12$ and its height is, say, $h$.
$h$/$6$ = $12$/$8$
which gives $h$ = $9$.
Now the volume can be found using the formula stated in the problem.