Question

A cylindrical container of radius \[6\text{ }cm\] and height \[15\text{ }cm\] is filled with ice-cream. The whole ice-cream has to be distributed to ten children in equal cones with hemispherical tops. If the height of the conical portion is four times the radius of its base, find the radius of the ice-cream cone.

Answer 1

Hint: To find the radius of the ice-cream cone we first need to find the amount of ice-cream by volume filled in the cylindrical container and to find that, we need to find the volume of the cylindrical container. Once the volume of the cylinder is obtained so will be the amount of ice-cream in volume. After that we need to find the volume of the ice-cream cone along with the ice-cream on the top in the form of hemisphere which means we need to find the sum of volume of the hemisphere and cone. After that we need to equate the sum of the volume of a single cone to the volume of ice-cream obtained by a single child (dividing the total volume by ten children).

Formula used:
Hence, the formula of volume used for each part is:
Volume of the cylindrical container \[=\pi {{R}^{2}}H\]
Volume of the cone \[=\dfrac{1}{3}\pi {{r}^{2}}h\]
Volume of a hemisphere \[=\dfrac{2}{3}\pi {{r}^{3}}\]
where \[R\] is the radius of the cylinder in centimeters and \[H\] is the height of the cylinder in centimeters and \[r\] is the radius of the cone and hemisphere and \[h\] is the height of the cone.

Complete step-by-step answer:
Now to find the volume of ice-cream each child get we need to divide the volume of the container by the number of children and i.e.
Each child receives an ice-cream volume of \[\dfrac{\text{Volum}{{\text{e}}_{\text{cylindrical container}}}}{\text{Number of children}}\]
\[=\dfrac{\pi {{R}^{2}}H}{\text{10}}\]
\[=\dfrac{\pi {{\left( 6 \right)}^{2}}15}{\text{10}}\]
\[=\dfrac{\pi \times 36\times 15}{\text{10}}\]
\[=169.56\approx 170c{{m}^{3}}\]
Now each child receives and ice-cream volume of \[170c{{m}^{3}}\]. Hence, equating the sum of the volume of cone and hemisphere of the ice-cream with the volume of ice-cream received by everyone we get:
\[170c{{m}^{3}}=\dfrac{1}{3}\pi {{r}^{2}}h+\dfrac{2}{3}\pi {{r}^{3}}\]
According to the question given, the height of the cone is equal to four times of the radius thereby, the height of the cone is \[h=r\]. Hence, the radius of the ice-cream cone is:
\[170c{{m}^{3}}=\dfrac{1}{3}\pi {{r}^{2}}\left( 4r \right)+\dfrac{2}{3}\pi {{r}^{3}}\]
\[170c{{m}^{3}}=\dfrac{1}{3}\left[ \pi {{r}^{2}}\left( 4r \right)+2\pi {{r}^{3}} \right]\]
Transforming the 3 in the denominator to the other side
\[170c{{m}^{3}}\times 3=\left[ \pi {{r}^{2}}\left( 4r \right)+2\pi {{r}^{3}} \right]\]
\[510c{{m}^{3}}=\left[ \pi {{r}^{2}}\left( 4r \right)+2\pi {{r}^{3}} \right]\]
\[510c{{m}^{3}}=\left[ 4\pi {{r}^{3}}+2\pi {{r}^{3}} \right]\]
Taking \[2\pi\] common and changing it to the other side
\[\dfrac{510c{{m}^{3}}}{2\pi }=\left[ 2{{r}^{3}}+{{r}^{3}} \right]\]
\[\dfrac{510c{{m}^{3}}}{2\times 3\pi }={{r}^{3}}\]
\[\sqrt[3]{\dfrac{510c{{m}^{3}}}{2\times 3\pi }}=r\]
\[\sqrt[3]{27.07}=r\]
Approximating the value of the cube root we get:
\[\sqrt[3]{27.07}\approx \sqrt[3]{27}=r\]
\[r=3cm\]

\[\therefore \] The radius of the ice-cream cone is \[3cm\].

Note: Students may be wrong when finding the radius of the cone as the sum of the volume of the cone and hemisphere is equal to the volume of one child’s ice-cream cone volume and not the volume of the total ice-cream contained. Hence, first divide the volume of the container by the total number of children and then equate it with the volume of a single ice-cream cone (sum of vol. of cone and vol. of hemisphere).