Question

One side of the cube box is 0.8 meters. How will it cost to paint the outer surface of the box at a rate of 150 rupees per square meter?

Answer 1

Hint: Cube is a three-dimensional figure/ shape, whose length, breadth and height are equal to each other in length.
The angle between two adjacent sides is $90^\circ $.

Total surface area of any cube = 6 x side x side sq. units
Total costs = Total area $ \times $Rate per sq. units

Complete step by step solution:
Step 1
Painting the cubical box means covering the surface area of all faces.
Express the relation of total surface area of cube and its side
\[{\rm{A = 6}} \times {{\rm{a}}^2}\]
Here, A is the total surface area of the cube and a is the length of the side of the cube.

Step 2
Substitute 0.8 m for side of cube as given in the question
\[\begin{array}{l}
{\rm{A = 6}} \times {\rm{0}}{\rm{.8m}} \times {\rm{0}}{\rm{.8m}}\\
{\rm{A = 3}}{\rm{.84}}{{\rm{m}}^2}
\end{array}\]
Here, A comes out to be 3.84 m$^2$.

Step 3
Express the relation of total cost of painting and area of the cube
\[{\rm{C = A}} \times {\rm{R}}\]
Here, C is the total cost of painting, A is the total area of the cube and R is the rate of painting per square meter.

Step 4
Substitute 3.84 as value of A and 150 as value of R as given in the question
\[\begin{array}{l}
{\rm{C = 3}}{\rm{.84}} \times 150\\
{\rm{C = 576 Rupees}}
\end{array}\]
C comes out to be 576 rupees.

Therefore, The cost to paint the outer surface of the box is 576 Rupees.

Additional information: Cuboid is another three-dimensional shape, whose length and breadth are not equal to each other.

Total area of cuboid having length ‘l ‘ breadth ‘b’ and height ‘h’ is \[2({\rm{l}} \times {\rm{b}} \times {\rm{h}})\]. Its lateral surface area can be calculated by using \[2{\rm{h}}({\rm{l}} \times {\rm{b}})\].

Note:
In these types of questions it is also common to ask its volume, its curved surface area, length of its diagonal etc. These can be calculated using following formulas
\[\begin{array}{l}
{\text{Curved surface area of a cube = 4 }} \times {\rm{(side)}} \times {\rm{(side)}}\\
{\text{Volume of the cube = (side)}} \times ({\rm{side}}) \times ({\rm{side}})\\
{\text{length of the diagonal of a cube = }}\sqrt 3 \times ({\rm{side}})
\end{array}\]