Question

A rectangular piece of metal with sides 24 cm and 18 cm costs 1,080 rupees. How much will a square piece of the same metal of side 16 cm costs if the cost of the metal is proportional to its area.
A) 840 rupees
B) 640 rupees
C) 720 rupees
D) 900 rupees

Answer 1

Hint: A rectangle is a four-sided 2-Dimensional figure in which opposite sides are equal in length. A square is also a four-sided 2-Dimensional figure in which all sides are of equal length.
A given Rectangle is formed by metal, which means the area of the rectangle will be the area of the metal piece.
If A is proportional to B means the factor by which A is increased B will also increase, it is also called A is directly proportional to B. The word proportional is denoted by a mathematical symbol \['\alpha '\] , the statement can also be written as:
$A{\text{ }}\alpha \,B$
When the proportional sign is replaced by $' = '$ a proportionality constant is multiplied. The statement can also be written as:
$A = kB$
Here k is proportionality constant.

Complete step-by-step answer:
Step 1: Find the area of the rectangle.

             Figure: Rectangle
The Given length of the rectangle = 24 cm
The breadth of the rectangle = 18 cm
Thus, the area of the rectangle is the product of the length and breadth of the rectangle.
Area of rectangle = length $ \times $ breadth
                                = 24 $ \times $ 18
                                = 432 sq. units
The given rectangle is made up by metal piece, thus the area of metal piece = 432 sq. units
                                                                                                                                                           …… (1)
Also, it is given that the rectangular piece of metal costs = 1,080 rupees. …… (2)
Step 2: Find proportionality constant.
It is given that the cost of metal is proportional to its area. The statement can also be written as:
Cost of the metal $\alpha $ area of metal
Cost of the metal = $k \times $ area of metal
Here k is proportionality constant.
Hence, the cost of metal used for rectangle = $k \times $ area of metal used for the rectangle.
From equation (1) and (2)
1080 = $k \times $432
$ \Rightarrow k = \dfrac{{1080}}{{432}}$
$\because k = 2.5$ …… (3)
Step 3: Find the area of the square.

                    Figure: Square
The given side of the square = 16 cm
Thus, the area of the square is the square of the length of its side.
Area of square = side $ \times $ side
                           = 16 $ \times $ 16
                           = 256 sq. units
The given square is made up by metal piece, thus the area of metal piece = 256 sq. units
                                                                                                                                                           …… (4)
Step 4: Find the costs of the square metal
We know from step 2 that:
Cost of the metal = $k \times $ area of metal
Here k is proportionality constant.
Hence, the cost of metal used for square = $k \times $ area of metal used for the square.
From equation (3) and (4)
The cost of metal used for square = 2.5 $ \times $ 256
                                                             = 640 rupees

Final answer: The cost of metal used for the square is 640 rupees. Thus, the correct option is (B).

Note: There is also one relation called inversely proportional. If A is indirectly proportional to B means the factor by which A is increased B will decrease. The statement can also be written as:
$A{\text{ }}\alpha {\text{ }}\dfrac{1}{B}$
Similarly, the proportional sign is replaced by $' = '$ a proportionality constant is multiplied. The statement can also be written as:
$A = k\dfrac{1}{B}$
Here k is proportionality constant.
In some questions related to the fencing of a rectangular or square field. The fencing is done along the boundary of the field. Thus, the perimeter of the field is calculated.
Hence, the perimeter of the rectangle is twice the sum of length and breadth of the rectangle.
The perimeter of rectangle = 2( length + breadth)
Hence, the perimeter of the square is four times the length of its side.
The perimeter of the square = 4 $ \times $ side.