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 really need help on this. I want the answer. Thanks!
the triangle has height 7√3/2
the cone base has a radius of 7
v = 1/3 pi r^2 h
plug and chug
To find the volume of the cone, first, let's find the height of the cone.
Since the equilateral triangle is revolved about an altitude, the altitude also becomes the height of the cone.
In an equilateral triangle, the altitude is related to the side length by the formula: height = (sqrt(3)/2) * side length.
Therefore, height = (sqrt(3)/2) * 14 = sqrt(3) * 7 ≈ 12.12 cm.
Now, let's find the radius of the cone. In an equilateral triangle, the altitude bisects the base, forming a right-angled triangle with half the base length as the hypotenuse.
Using Pythagoras' theorem, the base length of the right-angled triangle is calculated as: base length = sqrt((side length)^2 - (height)^2).
Substituting the values, base length = sqrt(14^2 - 12.12^2) ≈ sqrt(196 - 147.54) ≈ sqrt(48.46) ≈ 6.96 cm.
Now, we can calculate the volume of the cone using the formula: volume = (1/3) * π * (radius)^2 * height.
Substituting the values, volume = (1/3) * π * (6.96)^2 * 12.12 ≈ (1/3) * 3.14 * 48.38 * 12.12 ≈ 591.65 cubic centimeters.
Therefore, the number of cubic centimeters in the volume of the cone is approximately 592.
To find the volume of the cone formed by revolving the equilateral triangle, we can use the formula for the volume of a cone: V = (1/3) * π * r^2 * h, where V is the volume, π is a mathematical constant approximately equal to 3.14159, r is the radius of the base of the cone, and h is the height of the cone.
In this case, the equilateral triangle has sides of length 14 centimeters. The height of the cone is the altitude of the triangle, which is a line segment from one vertex perpendicular to the opposite side. Since an altitude of an equilateral triangle bisects the base, the height is the distance from the base to the midpoint of one side.
To find the height, we can use the properties of a 30-60-90 triangle. In an equilateral triangle, each angle is 60 degrees. Drawing an altitude creates two 30-60-90 right triangles. In a 30-60-90 triangle, the side lengths are related by the ratio 1 : sqrt(3) : 2. Since the side length of the equilateral triangle is 14 centimeters, the length of the altitude (height) is (sqrt(3)/2) times the side length.
Height = (sqrt(3)/2) * 14
Next, we need to find the radius of the base of the cone. The radius is half the length of a side of the equilateral triangle.
Radius = (1/2) * 14
Now, we can substitute the values for the height and radius into the volume formula:
V = (1/3) * π * (7)^2 * (14(sqrt(3)/2))
Simplifying this expression gives:
V ≈ 352 cubic centimeters
Therefore, the volume of the cone is approximately 352 cubic centimeters.