Rain equation of the parabola with vertex at origin given its directrix

1) y = (1/28) x²

2) x = (1/8) y²

3) y = (-1/16) x²

4) x = (-1/12) y²

Step-by-step explanation:

The vertex point is (0 , 0)

Directrix: y = k or x = k

The focus (a , b)

1) ∵ The directrix is y = -7 ⇒ below the vertex point by 7

   ∴ The distance between the vertex and the focus = 7

(The distance from the focus to the vertex = the distance from

 the vertex to the directrix)

∴ The focus is (0 , 7)

By using the rule of the distance

∵ (y - -7)² = (x - 0)² + (y - 7)²

∴ (y + 7)² = x² + (y - 7)²

∴ y² + 14y + 49 = x² + y² - 14y + 49

∴ 28y = x²

∴ y = (1/28) x²

2) ∵ The directrix is x = -2 ⇒ left the vertex point by 2

   ∴ The distance between the vertex and the focus = 2

(The distance from the focus to the vertex = the distance from

 the vertex to the directrix)

∴ The focus is (2 , 0)

By using the rule of the distance

∵ (x - -2)² = (x - 2)² + (y - 0)²

∴ (x + 2)² = (x - 2)² + y²

∴ x² + 4x + 4 = x² - 4x + 4 + y²

∴ 8x = y²

∴ x = (1/8) y²

3) ∵ The directrix is y = 4 ⇒ over the vertex point by 4

   ∴ The distance between the vertex and the focus = 4

(The distance from the focus to the vertex = the distance from

 the vertex to the directrix)

∴ The focus is (0 , -4)

By using the rule of the distance

∵ (y - 4)² = (x - 0)² + (y - -4)²

∴ (y - 4)² = x² + (y + 4)²

∴ y² - 8y + 16 = x² + y² + 8y + 16

∴ -16y = x²

∴ y = (-1/16) x²

4) ∵ The directrix is x = 3 ⇒ right to the vertex point by 3

   ∴ The distance between the vertex and the focus = 3

(The distance from the focus to the vertex = the distance from

 the vertex to the directrix)

∴ The focus is (-3 , 0)

By using the rule of the distance

∵ (x - 3)² = (x - -3)² + (y - 0)²

∴ (x - 3)² = (x + 3)² + y²

∴ x² - 6x + 9 = x² + 6x + 9 + y²

∴ -12x = y²

∴ x = (-1/12) y²

answer: ratio of volume of cone (with 3 times the height and equal diameter) to volume of cylinder is 2 to 1. the value of π is not used.

step-by-step explanation:

radius is diameter/2 = 4.

area of circle is π r^2.

volume of cylinder is height × area of base = hπr^2 = 3(4^2) π = 48π cubic inches.

not knowing the formula for volume of cone, i calculate it using calculus (which is for calculating things).

volume of cone is integral 0 to h area(y) dy. area(y) is π r(y)^2. r(y) runs r to zero as y runs zero to h, r(0) = r, r(h) = 0.

r(y) = r - r/h y

area(y) = π (r - r/h y)^2 = π (r^2 - 2r^2y/h + r^2y^2/h^2) = π r^2(1-2y/h+y^2/h^2)

antiderivative area(y) = π r^2 (y-y^2/h+y^3/(3h^2))

integral 0 to h = π r^2(h-h^2/h+h^3/(3h^2) - (0)) = π r^2(h - h + h/3) = π r^2(h/3) = π/3 r^2 h

volume of this cone is π/3(4^2)18 cubic inches.

= 6 16 π = 96π cubic inches.

cone has twice the volume, 96/48.

cone height is 6 times the cylinder.

ratio of volume formula: π r^2 cancels.

cone over cylinder ratio is 6h/3 to h, 2 to 1.