If the radius of a circle is decreased by 11% then the total decrease in the area of the circle is given as:
20.79%
Let's break down this problem about the change in the area of a circle when its radius changes. We are given that the radius of a circle is decreased by 11% and we need to find the total decrease in the area of the circle as a percentage.
Understanding Circle Area and Percentage Change
The area of a circle is calculated using the formula:
$$ A = \pi r^2 $$
where $A$ is the area and $r$ is the radius. When the radius changes, the area changes, but not linearly. Because the radius is squared in the formula, the area changes proportionally to the square of the change in the radius.
Calculating the New Radius
Let the original radius of the circle be $r_1$. The radius is decreased by 11%. This means the new radius, $r_2$, will be the original radius minus 11% of the original radius.
- Original radius = $r_1$
- Decrease in radius = 11% of $r_1 = 0.11 r_1$
- New radius, $r_2$ = $r_1 - 0.11 r_1 = (1 - 0.11) r_1 = 0.89 r_1$
So, the new radius is 89% of the original radius.
Calculating the Original and New Area
Using the area formula $A = \pi r^2$:
- Original area, $A_1 = \pi r_1^2$
- New area, $A_2 = \pi r_2^2$
Substitute the expression for $r_2$ into the formula for $A_2$:
$$ A_2 = \pi (0.89 r_1)^2 $$
Now, we calculate $(0.89)^2$:
$$ (0.89)^2 = 0.89 \times 0.89 = 0.7921 $$
So, the new area is:
$$ A_2 = \pi (0.7921 r_1^2) = 0.7921 (\pi r_1^2) $$
Since $A_1 = \pi r_1^2$, we can write the new area in terms of the original area:
$$ A_2 = 0.7921 A_1 $$
Calculating the Percentage Decrease in Area
The decrease in area is the original area minus the new area:
$$ \text{Decrease in area} = A_1 - A_2 $$
Substitute $A_2 = 0.7921 A_1$:
$$ \text{Decrease in area} = A_1 - 0.7921 A_1 = (1 - 0.7921) A_1 = 0.2079 A_1 $$
To find the percentage decrease in area, we divide the decrease in area by the original area and multiply by 100%:
$$ \text{Percentage decrease} = \frac{\text{Decrease in area}}{A_1} \times 100\% $$
$$ \text{Percentage decrease} = \frac{0.2079 A_1}{A_1} \times 100\% $$
The $A_1$ terms cancel out:
$$ \text{Percentage decrease} = 0.2079 \times 100\% = 20.79\% $$
So, the total decrease in the area of the circle is 20.79%.
Summary of Steps
Let's summarize the process:
- Determine the new radius after the percentage decrease.
- Calculate the original area using the original radius.
- Calculate the new area using the new radius.
- Find the absolute decrease in area (Original Area - New Area).
- Calculate the percentage decrease relative to the original area.
Let's look at the values:
| Measurement | Expression | Value (relative to original) |
|---|---|---|
| Original Radius ($r_1$) | $r_1$ | $1 r_1$ |
| New Radius ($r_2$) | $r_1 (1 - 0.11)$ | $0.89 r_1$ |
| Original Area ($A_1$) | $\pi r_1^2$ | $\pi r_1^2$ |
| New Area ($A_2$) | $\pi r_2^2 = \pi (0.89 r_1)^2$ | $0.7921 \pi r_1^2$ |
| Decrease in Area | $A_1 - A_2$ | $A_1 - 0.7921 A_1 = 0.2079 A_1$ |
| Percentage Decrease | $\frac{\text{Decrease}}{A_1} \times 100\%$ | $\frac{0.2079 A_1}{A_1} \times 100\% = 20.79\%$ |
The calculated percentage decrease in area is 20.79%.
Revision Table: Circle Area Percentage Change
| Concept | Formula / Idea | Notes |
|---|---|---|
| Area of Circle | $A = \pi r^2$ | Area depends on the square of the radius. |
| Percentage Decrease | $\frac{\text{Original Value} - \text{New Value}}{\text{Original Value}} \times 100\%$ | Always calculated relative to the original value. |
| Radius Decrease by $p\%$ | New Radius $= r (1 - p/100)$ | Radius is $(100-p)\%$ of original. |
| Area Change from Radius Change | $A_{new} = \pi (r_{new})^2$ | Substitute new radius into the area formula. |
Additional Information: Effect of Percentage Changes
When a quantity depends on the square of another quantity, like the area of a circle depending on the square of the radius, a percentage change in the base quantity results in a different percentage change in the derived quantity. If the radius changes by a factor $k$ (i.e., $r_{new} = k \cdot r_{original}$), the new area will be $A_{new} = \pi (k \cdot r_{original})^2 = \pi k^2 r_{original}^2 = k^2 \cdot A_{original}$.
In this problem, the radius is decreased by 11%, so the new radius is $100\% - 11\% = 89\%$ of the original radius. This means the factor $k = 0.89$.
The new area is $A_{new} = (0.89)^2 A_{original} = 0.7921 A_{original}$.
This means the new area is 79.21% of the original area. The decrease in area is $100\% - 79.21\% = 20.79\%$ of the original area.
A common mistake is to assume the area decrease is just twice the radius decrease (i.e., $2 \times 11\% = 22\%$), but this is an approximation only valid for very small percentage changes. The actual calculation involving squaring the factor is necessary for accuracy.
Important Questions from Plane Figures
If length of a rectangle is increased to its three times and breadth is decreased to its half, then the ratio of the area of given rectangle to the area of new rectangle is:
The width of the path around a square field is 4.5 m and its area is 105.75 m 2. Find the cost of fencing the field at the rate of Rs. 100 per meter.
What is the area of the square (in cm 2) whose vertices lie on a circle of radius 5 cm?
The circumcentre of an equilateral triangle is at a distance of 3.2 cm from the base of the triangle. What is the length (in cm) of each of its altitudes?
The perimeter of a circular lawn is 1232 m. There is 7 m wide path around the lawn. The area (in m 2) of the path is:
Take \(\left(\pi=\frac{22}{7}\right)\)
