Question

wooden bookshelf has external dimensions as follows: Height = 110 cm, depth = 25 cm, breadth = 85 cm. The thickness of the plank is 5 cm 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 \[c{{m}^{2}}\]. Find the total expenses required for polishing and painting the surface of the bookshelf.

A. Rs. 6275
B. Rs. 4340
C. Rs. 2525
D. Rs. 1935

Answer 1

- Hint:-Find the length and height of the open portion. Find the area of the bookshelf to be polished and its cost. Find the area to be painted and its cost. Total expense is the cost of polishing and the cost of painting..

Complete step-by-step answer: -

We have been given the external dimensions of a wooden bookshelf.
Height of the bookshelf = 110 cm
Depth of the bookshelf = 25 cm
Breadth of the bookshelf = 85 cm
Thickness of the plank = 5 cm
From the figure, we can say that the length and height of all open portions are the same. Now let us consider the length of open portions be \[{{l}_{1}}\] and the height of the open portion be \[{{h}_{1}}\].
Now let us find the height of the book shelf.
From the figure, we can say that,
Total height of bookshelf = \[3\times \] height of open portions + \[4\times \] thickness of the plank.
Thus we can write it as,
\[\begin{align}
  & 110=3{{h}_{1}}+4\times 5 \\
 & 110-20=3{{h}_{1}} \\
 & \therefore 3{{h}_{1}}=90 \\
 & {{h}_{1}}=\dfrac{90}{3}=30cm \\
\end{align}\]
Thus we got the height of an open portion as \[{{h}_{1}}=30cm\].
Now let us find the length of the bookshelf, from the figure we can say that,
length of bookshelf = length of open portion + \[2\times \] thickness of plank
\[\begin{align}
  & \therefore 85={{l}_{1}}+2\times 5 \\
 & \therefore {{l}_{1}}=85-10=75 \\
\end{align}\]
Thus we got the length of the open portion = \[{{l}_{1}}\] = 75 cm
It is said that the external faces are to be polished and the internal faces are to be painted.
Now the area of the open portion will be the area of the rectangle with length 75 cm and breadth 30 cm.
\[\therefore \]Area of bookshelf to be polished = total surface area of external bookshelf - \[3\times \] area of open position…..(1)
Now let us find the total surface area of the bookshelf. By looking in the figure, we can say that,
Total surface area (TSA) of external bookshelf = 2 (lb + bh + hl)
where l = 85 cm, b = 25 cm, h = 110 cm.
\[\therefore \]TSA of external portion \[=2\left( 85\times 25+25\times 110+110\times 85 \right)\]
\[\begin{align}
  & =2\left( 2125+2750+9350 \right) \\
 & =2\times 14225=28450c{{m}^{2}} \\
\end{align}\]
Now let us find area of open portion \[=3\times \left( {{h}_{1}}\times {{l}_{1}} \right)\]
\[\begin{align}
  & =3\times \left( 30\times 75 \right) \\
 & =3\times 2250=6750c{{m}^{2}} \\
\end{align}\]
Now let us substitute these values in equation (1).
\[\therefore \]Area of bookshelf to be polished = TSA of external bookshelf - \[3\times \] area of open portion.
Area of bookshelf to be polished = 28450 – 6750 = 21700 \[c{{m}^{2}}\]
It is said that the rate of polishing is 20 paise per \[c{{m}^{2}}\]and the rate of painting is 10 paise per \[c{{m}^{2}}\]
i.e. cost of painting 1 \[c{{m}^{2}}\] = 20 paise.
Thus cost of painting 21700 \[c{{m}^{2}}\] = \[21700\times 20=434000\] paise \[=\dfrac{434000}{100}=\] Rs. 4340
1 rupee = 100 paise
1 paise = \[\dfrac{1}{100}\] rupees
Now we need to paint the inner surface.
Thus total area to be painted = \[3\times \] (TSA of internal of shelf – area of open portion)……(2)
TSA of internal shelf = \[2\left( {{l}_{1}}{{b}_{1}}+{{b}_{1}}{{h}_{1}}+{{h}_{1}}{{l}_{1}} \right)\]
Here, \[{{l}_{1}}=75cm,{{b}_{1}}=25-5=20cm,{{h}_{1}}=30cm\]
\[\therefore \]TSA of internal shelf \[=2\left( 75\times 20+20\times 30+30\times 75 \right)\]
\[\begin{align}
  & =2\left( 2250+600+1500 \right) \\
 & =2\times 4320=8700c{{m}^{2}} \\
\end{align}\]
Area of the open portion = \[{{l}_{1}}\times {{h}_{1}}\] = \[75\times 30=2250c{{m}^{2}}\].
Now let us substitute these values in equation (2).
Total area to be painted = \[3\times \] (TSA of internal portion – area of open portion)
\[\therefore \]Total area to be painted \[=3\times \left( 8700-2250 \right)\]
\[\begin{align}
  & =3\times 6450 \\
 & =19350c{{m}^{2}} \\
\end{align}\]
It is said that cost of painting 1 \[c{{m}^{2}}\] = 10 paise
\[\therefore \]Cost of painting 19350 \[c{{m}^{2}}\] \[=19350\times 10\] \[=193500\] paise
\[=\dfrac{193500}{100}\] rupees = Rs. 1935.
Thus total expense = cost of polishing + cost of painting
= 4340 + 1935 = Rs. 6275
Thus we got the total expense as Rs. 6275.
Option A is the correct answer.

Note:
The wooden bookshelf is in the form of a cuboid. So the total surface area (TSA) is the area of all 6 faces of the cuboid. Thus TSA = 2lb + 2bh + 2lh \[=2\left( lb+bh+lh \right)\]. Then subtract the area of the open portion from the TSA to get the required answer.