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МатематикаВопрос:
Calculate the area of the shaded region
Ответ:
Let's analyze the geometry of the figure. The figure consists of a large semicircle with two smaller semicircles removed from its area.
1. Determine the radius of the large semicircle:
The diameter of the large semicircle is given as 16 cm. Therefore, the radius (R) is half of the diameter:
$$R = \frac{16 \text{ cm}}{2} = 8 \text{ cm}$$
2. Determine the radius of the smaller semicircles:
There are two identical semicircles. Their diameters form the diameter of the bigger semicircle, which is 16 cm. Therefore, the diameter of each smaller semicircle is half the diameter of the larger one: $$\frac{16}{2} = 8$$ cm. The radius (r) of each smaller semicircle is half of its diameter:
$$r = \frac{8 \text{ cm}}{2} = 4 \text{ cm}$$
3. Calculate the area of the large semicircle:
The area of a full circle is given by the formula $$A = \pi R^2$$. Since we have a semicircle, we take half of the area:
$$A_{\text{large}} = \frac{1}{2} \pi R^2 = \frac{1}{2} \pi (8 \text{ cm})^2 = \frac{1}{2} \pi (64 \text{ cm}^2) = 32\pi \text{ cm}^2$$
4. Calculate the area of one small semicircle:
Using the same logic as above, the area of one small semicircle is:
$$A_{\text{small}} = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (4 \text{ cm})^2 = \frac{1}{2} \pi (16 \text{ cm}^2) = 8\pi \text{ cm}^2$$
5. Calculate the total area of the two small semicircles:
Since there are two such semicircles, the total area of the two semicircles is:
$$2 × A_{\text{small}} = 2 × 8\pi \text{ cm}^2 = 16\pi \text{ cm}^2$$
6. Calculate the area of the shaded region:
The area of the shaded region is the area of the large semicircle minus the area of the two smaller semicircles:
$$A_{\text{shaded}} = A_{\text{large}} - 2A_{\text{small}} = 32\pi \text{ cm}^2 - 16\pi \text{ cm}^2 = 16\pi \text{ cm}^2$$
Answer: The area of the shaded region is $$16\pi \text{ cm}^2$$.
Using $$ \pi \approx 3.14 $$, we can approximate:
$$ A_{\text{shaded}} \approx 16 \cdot 3.14 \text{ cm}^2 = 50.24 \text{ cm}^2$$
Answer: $$16\pi \text{ cm}^2 \approx 50.24 \text{ cm}^2$$
1. Determine the radius of the large semicircle:
The diameter of the large semicircle is given as 16 cm. Therefore, the radius (R) is half of the diameter:
$$R = \frac{16 \text{ cm}}{2} = 8 \text{ cm}$$
2. Determine the radius of the smaller semicircles:
There are two identical semicircles. Their diameters form the diameter of the bigger semicircle, which is 16 cm. Therefore, the diameter of each smaller semicircle is half the diameter of the larger one: $$\frac{16}{2} = 8$$ cm. The radius (r) of each smaller semicircle is half of its diameter:
$$r = \frac{8 \text{ cm}}{2} = 4 \text{ cm}$$
3. Calculate the area of the large semicircle:
The area of a full circle is given by the formula $$A = \pi R^2$$. Since we have a semicircle, we take half of the area:
$$A_{\text{large}} = \frac{1}{2} \pi R^2 = \frac{1}{2} \pi (8 \text{ cm})^2 = \frac{1}{2} \pi (64 \text{ cm}^2) = 32\pi \text{ cm}^2$$
4. Calculate the area of one small semicircle:
Using the same logic as above, the area of one small semicircle is:
$$A_{\text{small}} = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (4 \text{ cm})^2 = \frac{1}{2} \pi (16 \text{ cm}^2) = 8\pi \text{ cm}^2$$
5. Calculate the total area of the two small semicircles:
Since there are two such semicircles, the total area of the two semicircles is:
$$2 × A_{\text{small}} = 2 × 8\pi \text{ cm}^2 = 16\pi \text{ cm}^2$$
6. Calculate the area of the shaded region:
The area of the shaded region is the area of the large semicircle minus the area of the two smaller semicircles:
$$A_{\text{shaded}} = A_{\text{large}} - 2A_{\text{small}} = 32\pi \text{ cm}^2 - 16\pi \text{ cm}^2 = 16\pi \text{ cm}^2$$
Answer: The area of the shaded region is $$16\pi \text{ cm}^2$$.
Using $$ \pi \approx 3.14 $$, we can approximate:
$$ A_{\text{shaded}} \approx 16 \cdot 3.14 \text{ cm}^2 = 50.24 \text{ cm}^2$$
Answer: $$16\pi \text{ cm}^2 \approx 50.24 \text{ cm}^2$$
