Question

BRILLIANT Q

A bowl is the shape of a hemisphere and made out of thick clay. The diameter of the outside bowl is 44.6 cm. And the diameter of the inside bowl is 41.6 cm.
a) If the bowl is filled up to the top, find how much soup could it hold?
b) Calculate the volume of clay that is needed to make a single bowl?
c) Six bowls are to be packed in a box side by side. The boxes dimensions are 22.5 cm by 90 cm by 135 cm. Sawdust packing is put in the box. What volume of the sawdust packing is needed?

Answer 1

The problem is all about volumes.

The volume of a hemisphere is defined as

`V=(2)/(3) \pi r^(3)`

According to the problem

`d_(outside)=44.6cm`

`d_(inside)=41.6cm`

Using the definition of radius

`r_(outside)=(44.6cm)/(2)=22.3cm`

`r_(inside)=(41.6cm)/(2)=20.8cm`

Now, the volume of the inside bowl is

`V_(inside)=(2)/(3)\pi(20.8cm) ^(3) \approx 5999.27 (3.14) \ cm^(3) \approx 18837.71 \ cm^(3)`

Therefore, the bowl can have 18,837.71 cubic centimeters of soup. (A)

The volume of the clay needed can be found with the differnece between hemispheres.

`V_(clay)=V_(outside) -V_(outside)`

`V_(clay)=(2)/(3)(3.14)(22.3cm)^(3)-18837.71cm^(3)= 23214.16 cm^(3)-18837.71cm^(3)= 4376.45 \ cm^(3)`

Therefore, we needed 4376.45 cubic centimeters of clay to make a single bowl. (B)

At last, the space that a single bowl occupies in the box is

`V_(outside)=23214.16 \ cm^(3)`

If we have 6 bowls, the total volume is

`V_(total)=6(23214.16 \ cm^(3) )= 139284.96 \ cm^(3)`

Now, we need to find the volume of the box

`V_(box)=22.5cm * 90cm * 135 cm = 273375cm^(3)`

The empty space between the box and the 6 bowls is

`V_(sawdust)=V_(box) - V_(total)=273375 cm^(3)-139284.96 cm^(3) =134090.04cm^(3)`

This empty space volume represents the sawdust packing needed, because the function of it is to fill the empty space.

Therefore, we need 134090.04 cubic centimeters of sawdust packing. (C)