Question

The sum of length, breadth, and height of a room is $ 19\,m $ . The length of the diagonal is $ 11\,m $ . The cost of painting the total surface area of the room at the rate of $ Rs.10 $ per $ {m^2} $ is:
a) $ Rs.\,240 $
b) $ Rs.\,2400 $
c) $ Rs.\,430 $
d) $ Rs.\,4200 $

Answer 1

Hint: In this question, we need to determine the total cost of painting the total surface area of the room at the rate of $ Rs.10 $ per $ {m^2} $ . We know the formula for diagonal of cuboid and the value if diagonal of the cuboid is given. So, by equating and evaluating we will get TSA of cuboidal room. Then multiplying it by $ 10 $ , as the rate of painting the room is $ Rs.10 $ per $ {m^2} $ , we will get the required cost.

Complete step-by-step answer:
It is given that the sum of length, breadth, and height of a room is $ 19\,m $ .
Then, we have,
 $ l + b + h = 19\,m $
It is also given that the length of the diagonal is $ 11\,m $ , therefore we have,
  $ D = 11\,m $
Now, we know that the diagonal of the cuboid is,
 $ D = \sqrt {{l^2} + {b^2} + {h^2}} $
Therefore, we have,
 $ \sqrt {{l^2} + {b^2} + {h^2}} = 11\,m $
Now, squaring the equation $ l + b + h = 19\,m $ on both sides, we have,
 $ {\left( {l + b + h} \right)^2} = 361\,m $
We know that, $ {\left( {a + b + c} \right)^2} = {a^2} + {b^2} + {c^2} + 2ab + 2bc + 2ca $
So, by applying the identity, we have,
 $ {l^2} + {b^2} + {h^2} + 2\left( {lb + bh + hl} \right) = 361 $
Squaring the equation $ \sqrt {{l^2} + {b^2} + {h^2}} = 11 $ on both sides, we have,
 $ {l^2} + {b^2} + {h^2} = 121 $
Applying the value of $ {l^2} + {b^2} + {h^2} = 121 $ in the equation $ {l^2} + {b^2} + {h^2} + 2\left( {lb + bh + hl} \right) = 361 $ , we have,
 $ 121 + 2\left( {lb + bh + hl} \right) = 361 $
 $ 2\left( {lb + bh + hl} \right) = 361 - 121 $
 $ 2\left( {lb + bh + hl} \right) = 240\,{m^2} $
we know that the total surface area of a cuboid $ = 2\left( {lb + bh + hl} \right) $
Therefore, total surface area of cuboidal room $ = 240\,{m^2} $
The cost of painting the total surface area of the room at the rate of $ Rs.10 $ per $ {m^2} $ .
Now, the cost of painting the total surface area of the room $ =\left( {10 \times 240} \right) $ $ = Rs.\,2400 $ .
Hence, option (b) $ Rs.\,2400 $ is the correct answer.
So, the correct answer is “Option b”.

Note: In this question, it is important to note that we need to be clear about the curved surface area and total surface area of the shapes. By which we can get an instant idea about solving the questions. Here also by using the given we have evaluated it into the form of total surface area of a cuboid by which we got the required solution.