An ice cream cone is 8 inches tall with a slant height of 10 inches . The opening of the cone is a circle.What is the diameter of the opening of the cone?
[Math] An ice cream cone is 8 inches tall and
circleseducationgeometry
Related Solutions
Extend the cone to its tip. Consider the following $2$ shapes:
Shape $1$: The cone, from the tip up to the place where it meets the second scoop from the bottom, together with the bottom scoop and the second scoop.
Shape $2$: The cone, from the tip up to the place where it meets the third scoop from the bottom, together with the second scoop from the bottom, and the third scoop from the bottom.
Shapes $1$ and $2$ are similar. Let $r$ be the scaling factor that gets us from the bottom scoop of Shape $1$ to the top scoop of Shape $1$. Then scaling again by the factor $r$ gets us to the top scoop of Shape $2$. And scaling twice more by the factor $r$ gets us to the top scoop in the diagram of the OP.
Thus $\frac{18}{8}=r^4$. It follows that $r^2=\frac{3}{2}$, and therefore the middle scoop has radius $(8)(3/2)$. Now that we have the radius, the volume is easy to write down.
Remark: We could do basically the same argument by drawing a bunch of similar triangles. But that would take longer, and anyway I wanted to give a pure scaling argument.
The argument above shows that the radius is indeed the geometric mean of $12$ and $18$. Precisely the same argument shows that if the bottom scoop has radius $a$ and the top scoop has radius $b$, then the middle scoop has radius $\sqrt{ab}$.
The same argument shows that if the bottom scoop as surface area $A$, and the top scoop has surface are $B$, then the middle scoop has surface area $\sqrt{AB}$.
The same argument shows that if the cone is not right-circular, again the middle scoop has radius $\sqrt{ab}$. The same applies to waffle cones that have the shape of an upside-down pyramid.
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.
Best Answer
Hint: draw a diagram and apply Pythagoras' Theorem - $\sqrt{10^2-8^2}= ....$ This gives you half the diameter of the cone. By the way, if $r$ is the radius of the ice cream then you can see that $6:r=8:10$, whence the radius of your ice cream is 7.5 inch.