The angles of a triangle are in the ratio 1:2:3. If its hypotenuse is 12cm, find the volume of the double cone formed by revolution of triangle about its hypotenuse.
Pls help me!
Sure, I'll be happy to help you out... with a side of humor, of course!
Well, to find the volume of the double cone formed by revolution of the triangle, we first need to find the dimensions of the cone. Since the angles of the triangle are in the ratio 1:2:3, let's assign them values of x, 2x, and 3x.
Now, let's use a bit of trigonometry to find the side lengths of the triangle. Using the sine function, we have:
sin(x) = opposite/hypotenuse
sin(x) = sqrt(1^2 + 2^2)/12
sin(x) = sqrt(5)/12
sin(2x) = sqrt(4^2 + 2^2)/12
sin(2x) = sqrt(20)/12 = sqrt(5)/6
sin(3x) = sqrt(9^2 + 2^2)/12
sin(3x) = sqrt(85)/12
Now that we have the side lengths, we can calculate the base radius (r) and height (h) of the cones formed by revolution. The radius (r) is equal to the base side length of the triangle, and the height (h) is equal to the hypotenuse. So:
r = sqrt(5)/12
h = 12
Now, let the volume of the double cone be V. The formula for the volume of a cone is V = (1/3) * pi * r^2 * h. Plugging in our values:
V = (1/3) * pi * (sqrt(5)/12)^2 * 12
V = (1/3) * pi * 5/144 * 12
V = (5/12) * pi
So, the volume of the double cone formed by revolution of the triangle about its hypotenuse is (5/12) times the value of pi.
To find the volume of the double cone formed by revolution of the triangle about its hypotenuse, we first need to find the lengths of the sides of the triangle.
Let's assume the angles of the triangle are x, 2x, and 3x. Since the sum of the angles in a triangle is 180 degrees, we can write the equation:
x + 2x + 3x = 180
6x = 180
x = 30
So the angles of the triangle are 30 degrees, 60 degrees, and 90 degrees.
Next, we need to find the lengths of the sides of the triangle. Since the hypotenuse is 12 cm, we can use trigonometry to find the other side lengths.
For the 30-degree angle, we have:
sin(30) = Opposite / Hypotenuse
sin(30) = Length of opposite side / 12 cm
Length of opposite side = sin(30) * 12 cm
Length of opposite side = 0.5 * 12 cm
Length of opposite side = 6 cm
Similarly, for the 60-degree angle, we have:
sin(60) = Length of opposite side / Hypotenuse
sin(60) = Length of opposite side / 12 cm
Length of opposite side = sin(60) * 12 cm
Length of opposite side = (√3 / 2) * 12 cm
Length of opposite side = 6√3 cm
Now, we have all the side lengths of the triangle:
Side opposite to the 30-degree angle: 6 cm
Side opposite to the 60-degree angle: 6√3 cm
Hypotenuse: 12 cm
To find the volume of the double cone, we can use the formula:
Volume = (1/3) * π * r^2 * h
The radius, r, is half the length of the side opposite to the 30-degree angle:
r = 6 cm / 2
r = 3 cm
The height, h, is the length of the side opposite to the 60-degree angle:
h = 6√3 cm
Plugging these values into the formula, we have:
Volume = (1/3) * π * (3 cm)^2 * 6√3 cm
Volume = (1/3) * π * 9 cm^2 * 6√3 cm
Volume = (1/3) * π * 54 cm^3 * √3 cm
Volume = 18π cm^3 * √3
Therefore, the volume of the double cone formed by revolution of the triangle about its hypotenuse is 18π cm^3 * √3.
To find the volume of the double cone formed by the revolution of the triangle, we first need to find the lengths of the two congruent cones that make up the double cone.
1. Find the lengths of the congruent cones:
Since the angles of the triangle are in the ratio 1:2:3, we can assume that the angles are x, 2x, and 3x (where x is a common factor).
We know that the sum of the angles in a triangle is 180 degrees, so we have: x + 2x + 3x = 180.
Simplifying the equation, we have: 6x = 180.
Dividing both sides by 6, we get: x = 30.
Therefore, the angles of the triangle are 30 degrees, 60 degrees, and 90 degrees.
Now, we can use trigonometry to find the lengths of the congruent cones:
- The base of each cone is the side opposite the 30 degrees angle, which is half of the hypotenuse (since the hypotenuse is opposite the 90 degrees angle in a right triangle). So, the base length of each cone is 12/2 = 6 cm.
- The height of each cone is the side opposite the 60 degrees angle (the side adjacent to the 30 degrees angle), which can be found using the sine function. The sine of 60 degrees is √3/2, so the height of each cone is (√3/2) * 6 = 6√3 cm.
2. Find the volume of each cone:
The volume of a cone can be found using the formula: V = (1/3) * π * r^2 * h, where r is the radius of the base and h is the height.
Since we have the base length (6 cm) and the height (6√3 cm) for each cone, we can find the volume of each cone:
V = (1/3) * π * (6/2)^2 * (6√3)
= (1/3) * π * 3^2 * (6√3)
= (1/3) * π * 9 * (6√3)
= 18π√3 cm^3.
3. Find the volume of the double cone:
Since the double cone is formed by two congruent cones, we need to double the volume of one cone to find the total volume:
Volume of double cone = 2 * 18π√3
= 36π√3 cm^3.
Therefore, the volume of the double cone formed by the revolution of the triangle about its hypotenuse is 36π√3 cm^3.
x + 2x + 3 x = 180
so
x = 30
we are talking about a 30, 60, 90 triangle
sides are 6 and 6 sqrt 3
bottom cone (I have the 6 side down)
hypotenuse is 6
length along axis = 6 sin 30 = 6/2 = 3
length perp to axis = 3 sqrt 3
volume of bottom cone
= (1/3) 3 (pi) (3 sqrt3)^2
= pi (9*3) = 27 pi
top cone
length along axis = 12-3 = 9
same 3 sqrt 3 perp to axis
Vol = (1/3)(9)(3 sqrt 3)^2
= 81 pi
total = (27+81)pi = 108 pi
so I have a cone