Solids [Surface Area & volume] — Case-Study Based Question

A warehouse is a large building which, in general, is used to store goods. The dimensions of a warehouse vary with resepct to the goods to be stored.

The dimensions of a particular warehouse are 88 m × 66 m × 5.5 m. Its owner wants to fill it completely with identical cubical cartons of maximum volume.

(i) What is the largest cube size that can fit perfectly ?

(ii) What is the volume of the warehouse ?

(iii) What is the volume of the cubical carton ?

(iv) What is the maximum number of cartons that can be stored in the warehouse ?

Answer

Given,

Dimensions of warehouse = 88 m × 66 m × 5.5 m

(i) Largest cube size that can fit = HCF of 88, 66 and 5.5

HCF of 88, 66, 5.5 = 5.5.

Hence, largest cube size that can fit = 5.5 m.

(ii) Volume of warehouse = l × b × h

= 88 × 66 × 5.5

= 31944 m³.

Hence, volume of warehouse = 31944 m³.

(iii) Size of a cubical carton = 5.5 m

Volume of cube = (side)³

= (5.5)³

= 166.375 m³.

Hence, volume of a cubical carton = 166.375 m³.

(iv) Maximum number of cartons that can be stored = $\dfrac{\text{Warehouse volume}}{\text{Cube volume}}$

= $\dfrac{31944}{166.375}$

= 192.

Hence, maximum number of cartons that can be stored in warehouse = 192.

Study the following picture of a study table made by Rohan for his SUPW project with match boxes of each dimension 6 cm by 4 cm by 1.5 cm.

Answer the following:

(i) How many match boxes are used to make the table ?

(ii) What is the floor are covered by the table ?

(iii) What is the surface area of the top of the table ?

(iv) What is the total volume of the table ?

Answer

Given,

Each matchbox size = 6 cm × 4 cm × 1.5 cm

(i) Total number of match boxes used to make table are:

Left support (leg) = 2 boxes

Middle span (the top) = 1 box

Right support(drawers) = 3 boxes

Total = 2 + 1 + 3 = 6 boxes.

Hence, total number of match boxes used to make the table = 6 boxes.

(ii) Area of one base = 6 × 4 = 24 cm²

Since there are two bases:

The area covered by left section : 6 × 1.5 = 9 cm²

The area covered by right section : 6 × 4 = 24 cm²

Total floor area covered by the table = 9 + 24 = 33 cm².

Hence, total floor area covered by the table = 33 cm².

(iii) Area of one base = 6 × 4 = 24 cm².

Since three distinct sections (left, middle, right) make up the table.

So,

Total surface area of table = 3 × 24

= 72 cm².

Hence, total surface area of the top of the table = 72 cm².

(iv) Since matchbox is in cuboid shape.

So, Volume of cuboid = l × b × h

= 6 × 4 × 1.5

= 36 cm³.

Total matchboxes required to make a table = 6

∴ Volume of the table = 6 × 36 = 216 cm³.

Hence, volume of the table = 216 cm³.