Question

A wooden bookshelf has external dimensions as follows. Height = 110cm Depth = 25cm and breadth = 85cm. The thickness of the plank is 5cm everywhere. The external faces are to be polished and the inner faces are to be painted. If the rate of polishing is 20 paise per $c{{m}^{2}}$ and the rate of painting is 10 paise per $c{{m}^{2}}$ . Then find the total expense required for polishing and painting the surface of the bookshelf.

Answer 1

Hint: Now first we will find the area of external faces. External faces consist of the top and bottom, the two side faces and the back face. Hence we will calculate their respective areas. Now once we have the total areas we can easily find the cost required to polish them.
Now consider the inner faces. Inner faces consist of the back face, the two side faces and the face of the planks. Hence we will again calculate the area of all these and hence find the total inner area of the shelf. Now once we have the total area we can again find the cost to paint them.

Complete step by step answer:
Now to find the area of the inner faces as well as outer faces we will calculate the area of particular rectangles. We know that area of rectangle is given by length × breadth
Now let us first consider the external faces.
Now in the given shelf the external faces include the top and bottom face, the two side face and the back face. Let us first calculate the area of each face.
Now the area of the top face.
Length of the top face is 85cm and the width is 25cm.
Now the area of the top face is $85\times 25=2125$
Hence the area of top face is $2125c{{m}^{2}}.....................\left( 1 \right)$
Now since the top face and bottom face are similar.
Area of bottom face is also $2125c{{m}^{2}}.....................\left( 2 \right)$
Now consider the side face.
Now the length of the side face is 110cm and the width of the side face is 25cm.
Now the area of side face will be $110\times 25=2750$
Hence the area of side face is $2750c{{m}^{2}}$
Hence the area of both the side face is $2\times 2750c{{m}^{2}}=5500c{{m}^{2}}$
Hence we have the area of both the side faces is $5550c{{m}^{2}}.........................\left( 3 \right)$
Now consider the back face.
The length of back face is 110cm and width is 85cm
Now the area of back face is $110\times 85=9350$
Hence the area of back face is $9350c{{m}^{2}}.............................\left( 4 \right)$
Hence from equation (1), equation (2), equation (3) and equation (4) we get
Total external surface is $9350+5550+2125+2125=19150c{{m}^{2}}$
Now the cost of polishing is 20 paise per $c{{m}^{2}}$
Hence the total cost of polishing will be $19150\times 20=383000$ paise.
Now 100 paise means 1 Rupee.
Hence 383000 paise means 3830 Rs
Hence the cost of polishing is 3830 Rs ……………………. (5).
Now consider the inner face.
Now there are three shelves in the bookshelf which are similar.
Hence instead of finding the area of all three planks we will just find the area of one plank and multiply it by 3 to get the area of planks.
Now consider the shelf.
Now let us calculate the length and breadth of plank.
Now each plank is of length 85 – 5 – 5 = 75cm
And breadth of each plank is 25cm
Hence we get area of each plank is $25\times 75=1875c{{m}^{2}}$
Hence the area of 3 such planks will be \[3\times 1875c{{m}^{2}}=5625c{{m}^{2}}................\left( 6 \right)\]
Now let us calculate the inner back face of the shelf.
Now the length of the back face is 85cm
Inner length of the back face is 85 – 5 – 5 = 75cm.
Now the width of back face is 110 – 5 – 5 – 5 – 5 = 90cm
Hence the area of inner back shelf is $90\times 75=6750c{{m}^{2}}..........................\left( 7 \right)$
Now consider the side face of the shelf.
The length and breadth of side face a=is 110 and 25 respectively.
Now length of inner side is 110 – 5 – 5 – 5 – 5 = 90cm
The width of the inner side remains the same as 25cm.
Hence the length area of inner side is $25\times 90=2250c{{m}^{2}}$
Now since there are two such side faces we get total area as $2\times 2250=4500.....................\left( 8 \right)$
Hence from equation (6), equation (7) and equation (8) we get the total inner surface area is
$4500+6750+5625=16875c{{m}^{2}}$
Now the rate of painting is 10 paise per $c{{m}^{2}}$
Hence the cost of painting $16875c{{m}^{2}}$ is 168750 paise.
Now since 100 paise is 1 Rs.
We get 168750 paise which is 1687.5 Rs.
Hence we have the total cost of painting is 1687.50Rs. …………………….. (9)
Hence from equation (5) and equation (9) we get.
Total cost is 1687.50 + 3830 = 5517.5 Rs.

Note: Now note that when calculating length and breadth check the dimensions correctly. Also while calculating the inner area do not forget to subtract thickness accordingly. Hence while calculating, study the diagram thoroughly so that no point is missed.