Question

A cylindrical container of radius 7cm and height 30.625cm is filled with ice cream. The whole ice cream has to be distributed to 15 children in equal cones with hemispherical tops. If the height of the conical portion is five times the radius of the base, then find the radius of the ice cream cone.

Answer 1

Therefore the radius of the cone is = 3.5 cm.

Step-by-step explanation:

Given, a cylinder container of radius 7 cm and height 30.625 cm is filled with ice cream.

Here r= 7 cm

and h = 30.625 cm

Therefore the volume of the cylinder is = $\pi r^2 h$

                                                                 $=(\pi \times 7^2\times 30.625)$  cm³

The volume of whole ice cream = The volume of the cylinder

                                                      $=(\pi \times 7^2\times 30.625)$   cm³

Let the radius of the cone be x cm.

Since, the height of the conical portion is 5 times the radius.

Then the height of the cone = 5 x cm.

Then the volume of each cone is = $\frac{1}{3} \pi r^2h$

                                                    $=(\frac{1}{3}\pi \times x^2 \times 5x)$ cm³

                                                    $=\frac{5}{3}\pi x^3$ cm³

Since , the whole ice cream has to be distributed to 15 children in equal with  hemisphere tops.

The radius of the hemisphere is = The radius of the cone =x

The volume of each hemisphere is $=\frac{2}{3}\pi r^3$

                                                          $=(\frac{2}{3}\pi x^3)$   cm³

The volume of ice cream in each cone is $=(\frac{5}{3}\pi x^3+\frac{2}{3}\pi x^3)$cm³

Therefore the total volume of the ice cream

$=(\frac{5}{3}\pi x^3+\frac{2}{3}\pi x^3)\times 15$ cm³

According to the problem,

$(\frac{5}{3}\pi x^3+\frac{2}{3}\pi x^3)\times 15= (\pi \times 7^2\times 30.625)$

$\Rightarrow \frac{7}{3}\pi x^3 \times 15=(\pi \times 7^2\times 30.625)$

$\Rightarrow x^3 =\frac{\pi \times 7^2\times 30.625\times 3}{7\pi \times 15}$

$\Rightarrow x^3 = 42.875$

$\Rightarrow x^3= \sqrt[3]{42.875}$

$\Rightarrow x =3.5$

Therefore the radius of the cone is = 3.5 cm.