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1 answer:
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Answer:
Step-by-step explanation:
step 1
Calculate the area of the shaded portion
step 2
Calculate the area of the unshaded portion
step 3
Find the difference of the areas
1)
Area of largest circle - 2 * Area of one smaller circle = Area of the shaded region
AE = diameter of large circle = 48cm
radius of larger circle = diameter / 2 = 48cm / 2 = 24cm
4 circles fit across the diameter of the circle, so the diameter of the larger circle = 4 * diameter of the smaller circle
diameter of larger circle = 48cm = 4 * diameter of the smaller circle
diameter of the smaller circle = 48cm / 4 = 12cm
radius of smaller circle = diameter / 2 = 12cm / 2 = 6cm
Area of a circle = pi * r^2
Now plug the circle area equation into the first equation:
![A_{shaded}=A_{l} - 2*A_{s}\\\\A_{shaded}=[\pi (r_{l})^{2}]-2*[\pi (r_{s})^{2}]\\\\A_{shaded}=[\pi (48cm)^{2}]-2*[\pi (6cm)^{2}]\\\\A_{shaded}=2304\pi-72\pi\\\\Area\ of\ shaded\ region\ is\ 2232\pi.](https://tex.z-dn.net/?f=A_%7Bshaded%7D%3DA_%7Bl%7D%20-%202%2AA_%7Bs%7D%5C%5C%5C%5CA_%7Bshaded%7D%3D%5B%5Cpi%20%28r_%7Bl%7D%29%5E%7B2%7D%5D-2%2A%5B%5Cpi%20%28r_%7Bs%7D%29%5E%7B2%7D%5D%5C%5C%5C%5CA_%7Bshaded%7D%3D%5B%5Cpi%20%2848cm%29%5E%7B2%7D%5D-2%2A%5B%5Cpi%20%286cm%29%5E%7B2%7D%5D%5C%5C%5C%5CA_%7Bshaded%7D%3D2304%5Cpi-72%5Cpi%5C%5C%5C%5CArea%5C%20of%5C%20shaded%5C%20region%5C%20is%5C%202232%5Cpi.)
2)
Area of the shaded region = 2/7 * Area of the smaller circle
Area of the unshaded region = Area of larger circle + Area of smaller circle - Area of shaded region * 2
![A_{unshaded}=[\pi (r_{1})^{2}]+[\pi (r_{2})^{2}]-2*[\pi (r_{2})^{2}]*\frac{2}{7}\\\\A_{unshaded}=[\pi (10cm)^{2}]+[\pi (7cm)^{2}] -\frac{4}{7}[\pi (7cm)^{2}]\\\\A_{unshaded}=100\pi\ cm^{2}+49\pi\ cm^{2}-\frac{4*49\pi\ cm^{2}}{7}\\\\A_{unshaded}=149\pi\ cm^{2}-(4*7*\pi\ cm^{2})\\\\A_{unshaded}=149\pi\ cm^{2}-28\pi\ cm^{2}\\\\\\A_{unshaded}=121\pi\ cm^{2}](https://tex.z-dn.net/?f=A_%7Bunshaded%7D%3D%5B%5Cpi%20%28r_%7B1%7D%29%5E%7B2%7D%5D%2B%5B%5Cpi%20%28r_%7B2%7D%29%5E%7B2%7D%5D-2%2A%5B%5Cpi%20%28r_%7B2%7D%29%5E%7B2%7D%5D%2A%5Cfrac%7B2%7D%7B7%7D%5C%5C%5C%5CA_%7Bunshaded%7D%3D%5B%5Cpi%20%2810cm%29%5E%7B2%7D%5D%2B%5B%5Cpi%20%287cm%29%5E%7B2%7D%5D%20-%5Cfrac%7B4%7D%7B7%7D%5B%5Cpi%20%287cm%29%5E%7B2%7D%5D%5C%5C%5C%5CA_%7Bunshaded%7D%3D100%5Cpi%5C%20cm%5E%7B2%7D%2B49%5Cpi%5C%20cm%5E%7B2%7D-%5Cfrac%7B4%2A49%5Cpi%5C%20cm%5E%7B2%7D%7D%7B7%7D%5C%5C%5C%5CA_%7Bunshaded%7D%3D149%5Cpi%5C%20cm%5E%7B2%7D-%284%2A7%2A%5Cpi%5C%20cm%5E%7B2%7D%29%5C%5C%5C%5CA_%7Bunshaded%7D%3D149%5Cpi%5C%20cm%5E%7B2%7D-28%5Cpi%5C%20cm%5E%7B2%7D%5C%5C%5C%5C%5C%5CA_%7Bunshaded%7D%3D121%5Cpi%5C%20cm%5E%7B2%7D)
Area of largest circle - 2 * Area of one smaller circle = Area of the shaded region
AE = diameter of large circle = 48cm
radius of larger circle = diameter / 2 = 48cm / 2 = 24cm
4 circles fit across the diameter of the circle, so the diameter of the larger circle = 4 * diameter of the smaller circle
diameter of larger circle = 48cm = 4 * diameter of the smaller circle
diameter of the smaller circle = 48cm / 4 = 12cm
radius of smaller circle = diameter / 2 = 12cm / 2 = 6cm
Area of a circle = pi * r^2
Now plug the circle area equation into the first equation:
2)
Area of the shaded region = 2/7 * Area of the smaller circle
Area of the unshaded region = Area of larger circle + Area of smaller circle - Area of shaded region * 2