Question

If the radius of the circle is decreased by \[10\% \] then there is ______ decrease in its area.
A.\[10\% \]
B.\[21\% \]
C.\[19\% \]
D.\[20\% \]

Answer 1

Hint: Here we will firstly find the original area of the circle. Then we will find the reduced radius then by using the reduced radius we will find the area of the new circle. Then we will find the reduced area by the difference between the original area and the area of the new circle. Then we will find the percentage of the area reduced.

Complete step-by-step answer:
Let r be the radius of the original circle and R be the radius of the new circle.
So, firstly we will find the area of the original circle. Therefore
Area of the original circle \[ = \pi {r^2}\]
Now we have to find out the reduced radius as it is given in the question that the radius is decreased by \[10\% \]. Therefore, we get
Radius of the new circle \[ = {\rm{original\, radius}} - {\rm{reduced\, radius}}\]
Radius of the new circle, \[R = r - \dfrac{{10r}}{{100}} = \dfrac{9}{{10}}r\]
Now we will find the area of the new circle, we get
Area of the new circle \[ = \pi {R^2}\]
By putting the value of the radius we get
Area of the new circle \[ = \pi {R^2} = \pi {\left( {\dfrac{9}{{10}}r} \right)^2} = \dfrac{{81}}{{100}}\pi {r^2}\]
Now we will find out the value of the reduced area to find out the percentage decrease in the area.
So, reduced area \[ = {\rm{area \,of \,original\, circle}} - {\rm{area\, of\, new\, circle}}\]
Reduced area \[ = \pi {r^2} - \dfrac{{81}}{{100}}\pi {r^2} = \dfrac{{19}}{{100}}\pi {r^2}\]
Now we have to find the percentage decrease in the area which will be equal to the ratio of the reduced area to the area of the original circle multiplied by 100 to get it in percentage. Therefore, we get
Percentage decrease in the area \[ = \dfrac{{{\rm{reduced\, area\,}}}}{{{\rm{area\, of\, original\, circle}}}} \times 100 = \dfrac{{\dfrac{{19}}{{100}}\pi {r^2}}}{{\pi {r^2}}} \times 100 = 19\% \]
Hence, \[19\% \] decreases in the area when the radius of the circle is decreased by \[10\% \].
So, option C is the correct option.

Note: We should note that what is given in the question its percentage decrease in the radius which means we have to subtract it from the original radius. We should know that area is the amount of surface covered by a shape in two dimensions. Area is generally measured in square units. Circumference is the similar concept as the perimeter of the shapes and both are generally measured in a single unit i.e. centimetre or meter.
Area of the circle\[ = \pi {r^2} = \dfrac{{\pi {d^2}}}{4}\], where d is the diameter of the circle.
Diameter of a circle is equal to two times the radius of the circle.
It is important to write the units of measurement with the values.