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.
how do we do this?!
What is the formula for area of cone?
or, more to the point, the volume of the cone is
1/3 pi r^2 h
you can figure r and h easily enough.
To find the volume of the cone formed by revolving the equilateral triangle, we need to follow these steps:
1. Start by finding the altitude of the equilateral triangle. In an equilateral triangle, each altitude is also the perpendicular bisector of a side. Therefore, by drawing the altitude, we create two right triangles, each with a hypotenuse of 14 cm and a base of 7 cm (half the side length of the triangle).
2. Using the Pythagorean theorem, we can find the height (altitude) of each right triangle. Let's call it 'h'. The equation will be: h^2 + 7^2 = 14^2.
Solving the equation: h^2 + 49 = 196
Subtracting 49 from both sides: h^2 = 147
Taking the square root to find 'h': h = sqrt(147)
Using a calculator, we find: h ≈ 12.12 cm.
3. The base of the cone will be the perimeter of the equilateral triangle; since all sides are equal, the base will be 14 + 14 + 14 = 42 cm.
4. Now we have all the components to calculate the volume of the cone. The formula for the volume of a cone is V = (1/3) * π * r^2 * h, where 'r' is the radius of the cone (which is half the base).
The radius 'r' = base/2 = 42/2 = 21 cm.
The height 'h' = 12.12 cm.
5. Plugging the values into the volume formula, we get:
V = (1/3) * π * (21^2) * 12.12
6. Using a calculator:
V ≈ 5400.02 cubic centimeters.
Rounding to the nearest whole number, the volume of the cone is approximately 5400 cubic centimeters.