An equilateral triangle of side 14 centimeters is revolved about an altitude to form a cone. What is the number of cubic centimeters in the volume of the cone? Express your answer to the nearest whole number, without units.
I just want the answer. Thanks! :)
Why did the equilateral triangle join the circus? Because it wanted to become a cone o-logist!
Now, let's calculate the volume of the cone. The formula for the volume of a cone is given by V = (1/3)πr²h, where r is the radius and h is the height.
In this case, since the equilateral triangle has side length 14 centimeters, each side is also the radius of the cone. So, r = 14 centimeters.
Now, let's find the height of the cone. The altitude of an equilateral triangle is given by h = √3/2 * a, where a is the side length. Therefore, h = √3/2 * 14.
Plugging these values into the volume formula, we get V = (1/3)π * (14)² * (√3/2 * 14).
Evaluating this expression, the volume of the cone is approximately 1081 cubic centimeters.
The volume of a cone can be calculated using the formula V = (1/3)πr^2h, where r is the radius of the base and h is the height of the cone.
In this case, we have an equilateral triangle with a side length of 14 centimeters. The altitude of an equilateral triangle is a line segment drawn from any vertex of the triangle perpendicular to the opposite side.
In an equilateral triangle, the altitude is also the median and the angle bisector, so it divides the triangle into two congruent right triangles. The base of each right triangle is half the length of one side of the equilateral triangle, which is 7 centimeters. The height of each triangle can be found using the Pythagorean theorem:
h^2 = (14^2) - (7^2)
h^2 = 196 - 49
h^2 = 147
h ≈ 12.124 centimeters
Since we are revolving the triangle about the altitude, the altitude becomes the height of the cone. Therefore, h = 12.124 centimeters.
To find the radius of the base, we can use the formula for the height of an equilateral triangle in terms of the side length:
h = (√3 / 2) * s
12.124 = (√3 / 2) * s
s = (2 * 12.124) / √3
s ≈ 14.002 centimeters
Since the triangle is approximately equilateral with a side length of 14.002 centimeters, we can approximate the radius of the base as r ≈ 14.002 / 2 ≈ 7.001 centimeters.
Now we can calculate the volume of the cone using the formula:
V = (1/3) * π * r^2 * h
V ≈ (1/3) * π * (7.001^2) * 12.124
V ≈ 269.22 cubic centimeters
Therefore, the volume of the cone is approximately 269 cubic centimeters.
To find the volume of the cone, we need to use the formula: V = (1/3)πr²h, where V is the volume, π is a mathematical constant (approximately equal to 3.14159), r is the radius of the base, and h is the height of the cone.
Given that the equilateral triangle has a side length of 14 centimeters, let's find the radius (r) and height (h) of the cone.
To find the radius, we can use a property of an equilateral triangle, where the radius is equal to (√3/6) times the side length. Thus, the radius (r) of the cone is: r = (√3/6) * 14 = (1.732/6) * 14 = 4.83 centimeters (approximately).
Now let's find the height (h) of the cone. The height is the altitude of the equilateral triangle, which divides it into two congruent 30-60-90 triangles. In a 30-60-90 triangle, the ratio of the sides is 1:√3:2. Since the side length of the triangle is 14 centimeters, the height of each 30-60-90 triangle is (√3/2) * 14 = (1.732/2) * 14 = 12.04 centimeters (approximately). Therefore, the height (h) of the cone is 2 * 12.04 = 24.08 centimeters (approximately).
Now we can calculate the volume of the cone using the formula mentioned above:
V = (1/3)πr²h
V = (1/3) * 3.14159 * (4.83)² * 24.08
V ≈ 737.76 cubic centimeters
Therefore, the volume of the cone is approximately 738 cubic centimeters.
Note that an equilateral triangle is a triangle where all lengths of the sides are equal.
If you revolve the triangle about its altitude (or the height), you'll generate a cone (just try to imagine or draw the figure). Therefore, the height of the cone is equal to the height of triangle, and its radius is equal to half of one side of the equilateral triangle, which is 14/2 = 7 cm (radius).
To get the height, use pythagorean theorem:
c^2 = a^2 + b^2
where
c = hypotenuse (in this case, 14 cm)
a = height of triangle (which is also cone height)
b = base (which is also the cone radius)
Substituting,
14^2 = a^2 + 7^2
a^2 = 14^2 - 7^2
a^2 = 147
a = 7*sqrt(3)
Finally, we get the volume of cone. Recall that the volume of cone is just
V = (1/3)*(pi)*(r^2)*h
Substituting,
V = (1/3)(3.14)(7^2)(7*sqrt(3))
Now solve for V. Units in cubic centimeters.
Hope this helps~ :)