Question

A wafer cone is completely filled with ice cream forms a hemispherical scoop, just covering the cone. The radius of the top of the cone, as well as the height of the cone are 7 cm each. Find the volume of the ice cream in it ( in $cm^3$).

Answer 1

Hint: In this question, we need to find the volume of the hemispherical scoop and the volume of ice cream occupied in the wafer cone. Then on adding volumes of hemispherical scoop and the volume of ice cream in the wafer cone we get the total volume of the ice cream.

Complete step-by-step answer:
Let us assume that the height of the cone as h and radius of the cone as r
Now from the given conditions in the question we have
\[\Rightarrow h=7,r=7\]

Now, let us assume the total volume of ice cream as V, volume of the hemispherical scoop as H and volume of the ice cream inside the cone as C
Now, we have the condition on the volumes as
\[\Rightarrow V=H+C\]
Now, let us first find the volume of the hemispherical scoop
As we already know that the volume of a hemisphere is given by the formula
\[\dfrac{2}{3}\pi {{r}^{3}}\]
As given in the question the value of the radius
\[\Rightarrow r=7\]
Now, the volume of the hemispherical scoop is given by
\[\Rightarrow H=\dfrac{2}{3}\pi {{r}^{3}}\]
Now, on substituting the respective values we get,
\[\Rightarrow H=\dfrac{2}{3}\times \dfrac{22}{7}\times {{7}^{3}}\]
Now, on cancelling the common terms and simplifying it further we get,
\[\Rightarrow H=\dfrac{44}{3}\times 49\]
Now, on further simplification we get,
\[\therefore H=\dfrac{2156}{3}c{{m}^{3}}\]
Now, let us find the volume of the ice cream inside the cone
As we already know that the volume of a cone is given by the formula
\[\dfrac{1}{3}\pi {{r}^{2}}h\]
As given in the question the value of the radius and height of the cone
\[\Rightarrow h=7,r=7\]
Now, the volume of the ice cream inside the cone is given by
\[\Rightarrow C=\dfrac{1}{3}\pi {{r}^{2}}h\]
Now, on substituting the respective values we get,
\[\Rightarrow C=\dfrac{1}{3}\times \dfrac{22}{7}\times {{7}^{2}}\times 7\]
Now, on cancelling the common terms and simplifying it further we get,
\[\Rightarrow C=\dfrac{22}{3}\times 49\]
Now, on further simplification we get,
\[\therefore C=\dfrac{1078}{3}c{{m}^{3}}\]
Now, the total volume of the ice cream is given by
\[\Rightarrow V=H+C\]
Now, on substituting the respective values we get,
\[\Rightarrow V=\dfrac{2156}{3}+\dfrac{1078}{3}\]
Now, this can also be written as
\[\Rightarrow V=\dfrac{3234}{3}\]
Now, on further simplification we get,
\[\therefore V=1078c{{m}^{3}}\]
Hence, the volume of the ice cream is 1078 $cm^3$.

Note: It is important to note that to get the total volume of the ice cream we need to add the volume of the scoop also other than the volume present inside the cone. It is also to be noted that the radius of the hemisphere will be the same as the cone because they have the same base.
Instead of finding the volumes separately and then adding them we can also find it by directly adding them while calculating and ten substitute the respective values. Both the methods give the same result.