Question

The side of a square is equal to the diameter of a circle. If the side and radius change at the same rate, then the ratio of the change of their areas is
(a) \[2:\pi \]
(b) \[\pi :1\]
(c) \[4:\pi \]
(d) \[1:2\]

Answer 1

Hint:
Here, we need to find the ratio of the area of the square to the area of the circle. First, we will find the areas of the square and the circle. Then, we will find the ratio of the areas by dividing them. Then, we will simplify the ratio using the given information to find the correct ratio.

Formula Used:
We will use the following formulas:
1) The area of a circle is given by the formula \[\pi {r^2}\], where \[r\] is the radius of the circle.
2) The area of a square is given by the formula \[{x^2}\], where \[x\] is the length of the square.
3) The radius of the circle is half of its diameter, that is \[r = \dfrac{d}{2}\].

Complete step by step solution:
Let the side of the square be \[a\], the diameter of the circle be \[d\], and the radius of the circle be \[r\].
The radius of the circle is half of its diameter, that is \[r = \dfrac{d}{2}\].
Multiplying both sides by 2, we get
\[ \Rightarrow d = 2r\]
Now, we will find the area of the square.
Substituting \[x = a\] in the formula \[{x^2}\], we get
Area of square \[ = {a^2}\]
It is given that the side of the square is equal to the diameter of the circle.
Therefore, we get
\[ \Rightarrow a = d\]
Substituting \[d = 2r\] in the equation, we get
\[ \Rightarrow a = 2r\]
Substituting \[a = 2r\] in the equation for area of the square, we get
Area of square \[ = {\left( {2r} \right)^2}\]
Simplifying the expression, we get
Area of square \[ = 4{r^2}\]
Now, we will find the area of the circle.
The area of a circle is given by the formula \[\pi {r^2}\], where \[r\] is the radius of the circle.
Therefore, we get
Area of circle \[ = \pi {r^2}\]
Finally, we will calculate the ratio of the area of the square to the area of the circle.
Dividing the area of the square by the area of the circle, we get
\[ \Rightarrow \] (Area of square \[ \div \] Area of circle)\[ = \dfrac{{4{r^2}}}{{\pi {r^2}}}\]
Simplifying the expression, we get
\[ \Rightarrow \] (Area of square \[ \div \] Area of circle) \[ = \dfrac{4}{\pi }\]
Therefore, we get the ratio of the area of the square to the area of the circle as \[4:\pi \].

Thus, the correct option is option (c).

Note:
A common mistake is to take the side of the square equal to the radius of the circle. This will result in the wrong area of the square, and hence, the incorrect ratio. Also, we need to find the area of square to area of circle and not the ratio of side of square to radius of circle. So, we need to carefully read the question.