the cross section of a solid in the form of a cone of height h, and a hemispherical bowl of radius r. the volume of the hemispherical portion is half that of the conical portion. calculate:
1)the ratio r:h
2)the vertical angle of the cone, correct to the nearest degree.
3)the total volume ,correct to three significant figures, of the solid, given that r=3.5cm [take pie=3.142
1) Well, to find the ratio r:h, we need to use the volume conditions given. Since the volume of the hemispherical portion is half that of the conical portion, we can write an equation:
1/2 * (1/2 * 4/3 * π * r^3) = 1/3 * π * (1/3 * h * r^2)
Simplifying this equation, we get:
2/3 * π * r^3 = 1/9 * π * h * r^2
Let's cancel out the common terms:
2/3 * r = 1/9 * h
To isolate r, we can multiply the equation by 9 and divide by 2:
r = 9/2 * h
Therefore, the ratio r:h = 9/2.
2) To find the vertical angle of the cone, we need to use the given information. We know that the volume of the hemispherical portion is half that of the conical portion. The volume of a cone is represented by (1/3 * π * r^2 * h), and the volume of a hemisphere is represented by (1/2 * 4/3 * π * r^3).
Setting up the equation:
(1/3 * π * r^2 * h) = 1/2 * (1/2 * 4/3 * π * r^3)
Simplifying:
(1/3 * π * r^2 * h) = 1/3 * π * r^3
Canceling out the common terms:
r^2 * h = r^3
Dividing both sides by r^2, we get:
h = r
Therefore, the vertical angle of the cone is the same as the angle of the hemisphere, which is approximately 180 degrees.
3) To calculate the total volume of the solid, we need to find the sum of the volumes of the cone and the hemisphere.
Volume of a cone = 1/3 * π * r^2 * h
Volume of a hemisphere = 1/2 * 4/3 * π * r^3
Given that r = 3.5 cm, let's calculate the volumes:
Volume of the cone = 1/3 * 3.142 * (3.5)^2 * 3.5 = 42.665 cm^3
Volume of the hemisphere = 1/2 * 4/3 * 3.142 * (3.5)^3 = 54.189 cm^3
Total volume = Volume of cone + Volume of hemisphere = 42.665 + 54.189 = 96.854 cm^3
Rounded to three significant figures, the total volume of the solid is approximately 96.9 cm^3.
Let's solve the problem step-by-step:
1) The ratio r:h:
Given that 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:
2 * r = h
Therefore, the ratio r:h is 1:2.
2) The vertical angle of the cone:
The vertical angle of a cone can be found by using the following formula:
tan(θ/2) = r / h
Using the given ratio r:h (1:2), we have:
tan(θ/2) = 1 / 2
Taking the inverse tangent of both sides:
θ/2 ≈ arctan(1/2)
Using a calculator, θ/2 is approximately 26.6 degrees.
Since we are looking for the full vertical angle of the cone (θ), we multiply θ/2 by 2:
θ ≈ 53.1 degrees
Therefore, the vertical angle of the cone is approximately 53.1 degrees.
3) The total volume of the solid:
To find the total volume of the solid, we need to sum the volume of the cone and the volume of the hemisphere:
Volume of the cone = (1/3) * π * r^2 * h
Volume of the hemisphere = (1/2) * (4/3) * π * r^3
Given that r = 3.5 cm and π = 3.142, we can calculate:
Volume of the cone = (1/3) * 3.142 * (3.5^2) * 2(3.5) = 27.26 cm^3 (rounded to two decimal places)
Volume of the hemisphere = (1/2) * (4/3) * 3.142 * (3.5^3) = 114.19 cm^3 (rounded to two decimal places)
Total volume = Volume of the cone + Volume of the hemisphere = 27.26 cm^3 + 114.19 cm^3 = 141.45 cm^3 (rounded to three significant figures)
Therefore, the total volume of the solid is approximately 141.45 cm^3.
To solve these questions, we need to understand the given information and the properties of cones and hemispheres.
1) The ratio r:h:
Let's assume the dimensions r (radius of the hemispherical bowl) and h (height of the cone) and then find the ratio.
The volume of the hemispherical portion is given as half the volume of the conical portion. The formula for the volume of a cone is V_cone = (1/3) * π * r^2 * h, and the formula for the volume of a hemisphere is V_hemisphere = (2/3) * π * r^3.
Given that V_hemisphere = (1/2) * V_cone, we can equate the two formulas:
(2/3) * π * r^3 = (1/3) * π * r^2 * h
Simplifying, we can cancel out the (1/3) and π terms:
2 * r^3 = r^2 * h
Dividing both sides by r^2:
2 * r = h
So, the ratio r : h is 1 : 2.
2) The vertical angle of the cone:
The vertical angle of a cone refers to the angle formed by the axis of the cone at its vertex. To find this angle, we can use trigonometry.
Let's assume the vertical angle of the cone as θ.
In a right triangle formed by the height h, the radius r, and the slant height l of the cone, we can use the sine function: sin(θ) = r / l.
The slant height l can be found using the Pythagorean theorem: l^2 = r^2 + h^2.
Substituting l^2 in the sine function equation:
sin(θ) = r / √(r^2 + h^2)
To find the angle θ, we can take the inverse sine (arcsine) of both sides:
θ = arcsin(r / √(r^2 + h^2))
Plugging in the values r = 3.5 cm and the ratio r : h = 1 : 2:
θ = arcsin(3.5 / √(3.5^2 + (2 * 3.5)^2))
Calculate the value of θ using a scientific calculator:
θ ≈ 30.96 degrees (rounded to the nearest degree).
Therefore, the vertical angle of the cone is approximately 31 degrees.
3) The total volume of the solid:
To find the total volume, we need to sum the volume of the cone and the volume of the hemisphere.
The volume of the cone is given by V_cone = (1/3) * π * r^2 * h, and the volume of the hemisphere is V_hemisphere = (2/3) * π * r^3.
Since we already know the ratio r : h = 1 : 2, we can substitute h = 2r into the volume formulas:
V_cone = (1/3) * π * r^2 * (2r) = (2/3) * π * r^3
The total volume is then:
V_total = V_cone + V_hemisphere
= (2/3) * π * r^3 + (2/3) * π * r^3
= (4/3) * π * r^3
Now, substitute the value of r = 3.5 cm and calculate the volume:
V_total = (4/3) * 3.142 * (3.5^3)
≈ 179.783 cm^3 (rounded to three significant figures)
Therefore, the total volume of the solid is approximately 179.783 cm^3.
we know that
2/3 pi r^3 = 1/2 (1/3 pi r^2 h), so
h = 2/3 r
so, r:h = 3:2
arctan(3/2) = 56.31 deg
not quite sure what you mean for part 2, but that should help
v = 2/3 pi r^3 + 1/3 pi r^2 h
= 2/3 pi r^3 + 1/3 pi r^2 (2/3 r)
= 8/9 pi r^3