Answer:-
Central angle , Ф/2π , A = Ф/2 r²
The ratio is 1/3 , V = 1/3 Bh
The equation of the circle in standard form is (x - 4)² + (k - 1)² = 25
Explanation:
* Lets revise the rules of the area of the sector of a circle
- The area of the sector which has a central angle Ф° is
(Ф°/360°) × πr², where 360° is the measure of the circle and r is
the radius of the circle
- The area of the sector which has a central angle Ф radians is
(1/2) r²Ф
* Lets complete the missing in the 1st picture
- The ratio of the sector's area A to the circle's area is equal to the
ratio of the central angle to the measure of a full rotation of the circle
- A full rotation of a circle is 2π. This proportion can written as
A/πr² = Ф/2π
- Multiply both sides by πr² to get A = Ф/2 r² where Ф is the measure
of the central angle and r is the radius of the circle
* Lets revise the rules of the volume of the prism and the volume
of the pyramid, where they have the same base and height
- The volume of the prism = area of the base × its height
- The volume of the pyramid = 1/3 × area of the base × its height
- From them the ratio of the volume of the pyramid to the volume
of the prism is 1/3
- The formula of the volume of the prism is V = Bh, where B is the
area of the base and h is the height, the formula of the volume
of the pyramid is V = 1/3 Bh
* Lets revise the standard form of the equation of a circle with
center (h , k) and radius r
- The equation is: (x - h)² + (y - k)² = r²
∴ x² - 2hx + h² + y² - 2ky + k² - r² = 0
∵ x² - 8x + y² - 2y - 8 = 0
- Lets equate the two equation
∴ x² - 2hx + h² + y² - 2ky + k² - r² = x² - 8x + y² - 2y - 8 = 0
∵ -2h = -8 ⇒ ÷ -2
∴ h = 4
∵ -2k = -2 ⇒ ÷ -2
∴ k = 1
∵ h² + k² - r² = -8
∴ (4)² + (1)² - r² = -8
∴ 16 + 1 - r² = -8
∴ 17 - r² = -8 ⇒ subtract 17 from both sides
∴ -r² = -15 × -1
∴ r² = 25
* Substitute the values of h , k , r in the equation of the standard
form of the circle
∴ (x - 4)² + (k - 1)² = 25
* The equation of the circle in standard form is (x - 4)² + (k - 1)² = 25