Question

The cost of diamond varies directly as the square of its weight. A diamond weight $ 10 $ gram cost $ 8000 $ Rs. If it breaks into two pieces whose weights are in the ratio of $ 3:2 $ . Then the loss incurred in rupees is,
A. $ 3840 $
B. $ 3960 $
C. $ 4040 $
D. $ 4160 $

Answer 1

Hint: First determine the proportionality constant which relates the cost and weight of the diamond. Then determine the weight of the two broken pieces of diamond and then the cost of the two pieces. The difference of the price of the 10 gram diamond and the sum of the cost of the two pieces will be the loss incurred.

Complete step-by-step answer:
The given information
The cost of the diamond is directly proportional to its weight. It can be written as
 $ \Rightarrow C = k \times {w^2} \cdots \left( 1 \right) $
The constant of proportionality is determined by substituting the weight $ w = 10 $ gram diamond and its cost $ C = 8000{\text{ Rs}} $ in equation (1), we get
 $
\Rightarrow 8000 = k \times {\left( {10} \right)^2} \\
\Rightarrow k = 80 \;
  $
The cost is related to its weight as
 $ C = 80 \times {w^2} \cdots \left( 2 \right) $
The diamond is broken down in two pieces in the ratio of $ 3:2 $ .
Let the weight of the two pieces of diamond be $ 3x $ and $ 2x $ .
The total weight is given equal to $ 10 $ grams.
Therefore,
 $ 3x + 2x = 10 \cdots \left( 3 \right) $
Solving equation (3) for, we get
 $
\Rightarrow 5x = 10 \\
\Rightarrow x = 2 \;
  $
The weight of the two pieces is $ 3 \times 2 = 6 $ gram and $ 2 \times 2 = 4 $ gram.
The cost of the 6 gram can be determined using equation (2) as,
 $
\Rightarrow {C_6} = 80 \times {\left( 6 \right)^2} \\
\Rightarrow {C_6} = 2880{\text{ Rs}} \;
  $
The cost of the 4 gram can be determined using equation (2) as,
 $
\Rightarrow {C_6} = 80 \times {\left( 4 \right)^2} \\
\Rightarrow {C_6} = 1280{\text{ Rs}} \;
  $
Total cost of the two pieces of the diamond,
 $ {C_T} = {C_6} + {C_4} \cdots \left( 4 \right) $
Substitute the value of and in equation (4) we get
 $
\Rightarrow {C_T} = 2880 + 1280 \\
\Rightarrow {C_T} = 4160{\text{ Rs}} \;
  $
The total cost of a $ 10 $ gram diamond is $ 8000 $ Rs. It is more than the cost of the two pieces of diamond.
The loss incurred is
 $
\Rightarrow L = 8000 - 4160 \\
\Rightarrow L = 3840{\text{ Rs}} \;
  $
So, the correct answer is “Option A”.

Note: It is important to note down one thing that the loss in weight occurred as the cost of the diamond is directly proportional to the square of its weight. If the cost is directly proportional to the weight then there will be no loss in the cost of the diamond on breakage.