The diagram represents the cross section of a solid in the form of a cone if height h, and a hemispherical bowl of radius r. The volume of the hemispherical portion is half that of the conical portion
Calculate:
(a). The ration r : h
(b). The vertical angle of the cone correct to the nearest degree;
(c). The total volume, correct to three significant figures, of the solid given that r = 3.5cm
( Take π = 3.142)
I need the answer as quick as possible pls
hemisphere: v = 2/3 πr^3
cone: v = 1/3 πr^2 h
since we have
2/3 πr^3 = 1/2 (1/3 πr^2 h)
that means that
(a) r/h = 1/4
(b) tan(θ/2) = r/h
(c) plug in your numbers using the formulas shown above
(a). To find the ratio r : h, we can start by comparing the volumes of the hemispherical portion and the conical portion. Since the volume of the hemispherical portion is half that of the conical portion, we can set up the following equation:
(1/2) * (4/3) * π * r^3 = (1/3) * π * r^2 * h
Simplifying the equation, we get:
2/3 * r = h
So, the ratio r : h is 2 : 3.
(b). To find the vertical angle of the cone, we can use the formula for the slant height of a cone:
L = √(r^2 + h^2)
Since we know the radius and height ratio (2 : 3), we can substitute these values into the formula:
L = √(r^2 + (2/3 * r)^2)
= √(r^2 + 4/9 * r^2)
= √(13/9 * r^2)
= (sqrt(13)/3) * r
Now, we can calculate the vertical angle (θ) using the formula:
sin(θ) = r / L
θ = arcsin(r / L)
θ = arcsin(r / ((sqrt(13)/3) * r))
Using a calculator, we find that θ ≈ 36 degrees.
(c). To find the total volume of the solid, we need to add the volumes of the conical portion and the hemispherical portion.
Volume of the conical portion (Vc) = (1/3) * π * r^2 * h
Volume of the hemispherical portion (Vh) = (1/2) * (4/3) * π * r^3
Substituting the given value r = 3.5 cm, and using π = 3.142, we can calculate the volumes:
Vc = (1/3) * π * (3.5 cm)^2 * (2/3) * (3.5 cm)
≈ 11.669 cm^3
Vh = (1/2) * (4/3) * π * (3.5 cm)^3
≈ 42.738 cm^3
Total volume (Vt) = Vc + Vh
Vt ≈ 11.669 cm^3 + 42.738 cm^3
≈ 54.407 cm^3
Therefore, the total volume of the solid is approximately 54.407 cm^3 (rounded to three significant figures).
To solve this problem, we'll use the formulas for the volume of a cone and the volume of a hemisphere.
(a) The ratio r : h:
The volume of the cone is given by V_cone = (1/3) * π * r^2 * h
The volume of the hemisphere is given by V_hemisphere = (1/2) * (4/3) * π * r^3
Since the volume of the hemisphere is half that of the cone, we can write:
V_hemisphere = (1/2) * V_cone
Substituting the formulas for V_cone and V_hemisphere:
(1/2) * (4/3) * π * r^3 = (1/3) * π * r^2 * h
Simplifying:
2 * r = 3 * h
The ratio r : h is therefore 2:3.
(b) The vertical angle of the cone:
We know that the cone is proportionally similar to the hemisphere. The vertical angle of a cone is twice the angle between the bottom edge and the apex. Since the hemisphere has a vertical angle of 180 degrees (a semicircle), the cone will have a vertical angle of 2 * 180 = 360 degrees.
(c) The total volume:
To calculate the total volume, we need to find the volumes of the cone and hemisphere and add them together.
Given r = 3.5 cm:
Volume of the cone: V_cone = (1/3) * π * (3.5)^2 * h
Volume of the hemisphere: V_hemisphere = (1/2) * (4/3) * π * (3.5)^3
To find the total volume:
Total Volume = V_cone + V_hemisphere
Substituting the formulas and using π = 3.142:
Total Volume = (1/3) * 3.142 * (3.5)^2 * h + (1/2) * (4/3) * 3.142 * (3.5)^3
Now you can calculate the total volume using the given height h.
To solve this problem, let's break it down step by step.
(a) The ratio r:h:
The volume of the hemispherical portion is half that of the conical portion, so we can set up an equation using the formulas for the volume of a cone and a hemisphere.
The volume of a cone is V_cone = (1/3) * π * r^2 * h
The volume of a hemisphere is V_hemisphere = (2/3) * π * r^3
Since the volume of the hemisphere is half that of the cone, we have:
V_hemisphere = (1/2) * V_cone
Substituting the formulas and simplifying:
(2/3) * π * r^3 = (1/2) * (1/3) * π * r^2 * h
Cancelling out π and simplifying further:
2 * r = h
So the ratio r:h is 2:1.
(b) The vertical angle of the cone:
To find the vertical angle of the cone, we can use trigonometry. The vertical angle of the cone is the angle between the vertical height (h) and the slant height of the cone.
Using the ratio r:h from part (a), we have r = (1/2) * h
Let's draw a right triangle to represent the cross-section of the cone. The height of the triangle is h, the base (which is also the radius of the hemisphere) is r, and the slant height of the cone is s.
Since r = (1/2) * h, we have r^2 = (1/4) * h^2
Using the Pythagorean theorem, we have h^2 + r^2 = s^2
Substituting r^2, we get h^2 + (1/4) * h^2 = s^2
Multiplying through by 4, we get 4h^2 + h^2 = 4s^2
Simplifying, we have 5h^2 = 4s^2
Taking the square root of both sides, we have √(5h^2) = 2s
Simplifying further, we have √5 * h = 2s
Dividing both sides by 2h, we have √5/2 = s/h
The vertical angle θ is the angle whose tangent is the ratio of the slant height (s) to the height (h):
tan(θ) = s/h
θ = arctan(s/h)
Since s/h = √5/2, we can calculate the angle:
θ = arctan(√5/2)
Using a calculator, this gives us θ ≈ 60 degrees (to the nearest degree).
(c) The total volume of the solid:
To find the total volume of the solid, we need to sum the volumes of the cone and hemisphere.
The volume of the cone is V_cone = (1/3) * π * r^2 * h
The volume of the hemisphere is V_hemisphere = (2/3) * π * r^3
Since we are given that r = 3.5 cm, we can substitute this value to calculate the volume.
V_cone = (1/3) * π * (3.5)^2 * h
V_hemisphere = (2/3) * π * (3.5)^3
The total volume is V_total = V_cone + V_hemisphere
Using your provided value of π ≈ 3.142 and substituting r = 3.5 cm, we can calculate V_total.