Question

The volume of a rectangular block of stone is $10360d{m^3}$. Its dimensions are in the ratio of 3:2:1. If the entire surface is polished at 2 paise per $d{m^2}$, then the total cost will be
A. Rs. 31.50
B. Rs. 63.00
C. Rs. 63.36
D. Rs. 31.68

Answer 1

Hint: First, let the dimensions be $3x$, $2x$ and $x$ and use the given condition to find the dimensions of the rectangular block. Use the dimensions of the rectangular block to find the total surface area of the block and then find the total area which is to be polished. Then, multiply the cost of per square unit to find the total cost of polishing.

Complete step by step Answer:

Let the common factor of the ratio be $x$.
Then the dimensions of the rectangular block will be $3x$, $2x$ and $x$
The rectangular block will be of the shape of a cuboid.
Now, we know that volume of a cuboid is given by the product of its dimensions.
We are given that the volume of the rectangular block is $10360d{m^3}$
Then, we have
$
  10360 = 3x\left( {2x} \right)\left( x \right) \\
  10360 = 6{x^3} \\
$
Divide the equation throughout by 6 and then take cube root of the resultant equation
$
  \dfrac{{10360}}{6} = {x^3} \\
   \Rightarrow x = \sqrt {\dfrac{{10360}}{6}} \\
   \Rightarrow x = 11.99 \\
   \Rightarrow x \approx 12dm \\
$
Next, we will find all the dimensions of the block.
The dimensions of the block will be;
$3\left( {12} \right) = 36dm$, $2\left( {12} \right) = 24dm$ and $12dm$
We want to find the cost of polishing if the rate is 2 paise per $d{m^2}$
The area covered for polishing will be the total surface area as all the faces are being polished.
TSA of a cuboid is given by $2\left( {lb + bh + hl} \right)$, where $l,b,h$ are the dimensions of the cuboid.
Then, TSA of the given rectangular block is
$2\left( {\left( {36} \right)\left( {24} \right) + \left( {24} \right)\left( {12} \right) + \left( {12} \right)\left( {36} \right)} \right)$
Which is equal to $2\left( {864 + 288 + 432} \right) = 3168d{m^2}$
Cost of polishing is 2 paise per $d{m^2}$, which is also equivalent to Rs. 0.02 per $d{m^2}$ because 1 rupee is equal to 100 paise.
Then, multiply the total area of the box with Rs. 0.02 to find the total cost.
$3168\left( {0.02} \right) = Rs.63.36$
Hence, option C is correct.

Note: We will find the area by calculating total surface area and not the curved surface area as CSA will cover only 4 faces of the cuboid but we have to paint the whole rectangular block. Students must remember the formula to do these types of questions.