a sector of a circle of radius 7 cm substance and angle of 270 degree at the centre of the circle is used to form a cone find the base radius of a cone and calculate the area of the base of the cone
A sector of a circle at radius 7cm subtending an angle of 270degree at the centre of the circle is used to form a cone ;find the base radius of the cone and calculate the area of the base of the cone
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pi * (.75*7)^2
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To find the base radius of the cone, we need to use the given information about the sector of the circle.
1. Since the angle at the center of the circle is 270 degrees, the fraction of the total circle is 270/360 = 3/4.
2. The circumference of the full circle is given by C = 2πr, where r is the radius of the circle.
3. From step 1, the fraction of the circumference of the circle covered by the sector is 3/4. So, the circumference of the sector is (3/4) * C.
4. Given that the radius of the circle is 7 cm, the circumference of the circle is C = 2π(7) = 14π cm. Therefore, the circumference of the sector is (3/4) * 14π cm.
5. The curved edge of the cone is formed by the circumference of the sector, so it would be the slant height of the cone.
6. The curved edge (slant height) of the cone is equal to the circumference of the sector, which is (3/4) * 14π cm.
7. The base of the cone is a circle, and the circumference of the base is equal to the circumference of the sector. Therefore, the base circumference of the cone is (3/4) * 14π cm.
8. The formula for the circumference of a circle is C = 2πr. Since we want to find the base radius of the cone, we can set up the equation (3/4) * 14π = 2πr and solve for r.
(3/4) * 14π = 2πr
(3/4) * 14 = 2r
r = (3/4) * 14 * (1/2)
r = 21 cm
So, the base radius of the cone is 21 cm.
Now, let's calculate the area of the base of the cone using the formula for the area of a circle.
1. The formula for the area of a circle is A = πr^2, where r is the radius of the circle (in this case, the base radius of the cone).
2. The base radius of the cone is 21 cm. Therefore, the area of the base of the cone is A = π(21^2) cm^2.
A = 441π cm^2
Therefore, the area of the base of the cone is 441π cm^2.