To find out how many times greater the area of a circle with a diameter of 200 mm is than that of a circle with a diameter of 40 mm, we first need to calculate the areas of both circles.
The formula for the area \( A \) of a circle is given by:
\[ A = \pi r^2 \]
where \( r \) is the radius of the circle.
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For the circle with a diameter of 200 mm:
\[ r = \frac{200 , \text{mm}}{2} = 100 , \text{mm} \] \[ A_1 = \pi (100 , \text{mm})^2 = \pi \times 10000 , \text{mm}^2 = 10000\pi , \text{mm}^2 \]
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For the circle with a diameter of 40 mm:
\[ r = \frac{40 , \text{mm}}{2} = 20 , \text{mm} \] \[ A_2 = \pi (20 , \text{mm})^2 = \pi \times 400 , \text{mm}^2 = 400\pi , \text{mm}^2 \]
Now, we find the ratio of the areas:
\[ \text{Ratio} = \frac{A_1}{A_2} = \frac{10000\pi}{400\pi} \]
The \( \pi \) cancels out:
\[ \text{Ratio} = \frac{10000}{400} = 25 \]
Therefore, the area of the circle with a diameter of 200 mm is approximately 25 times greater than the area of the circle with a diameter of 40 mm.