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 have tried solving this but it is hard how to imagine it...is the side like 14 in????
622
AoPS Answer Hackers, could you please not post answers here? AoPS has message boards you can use if in trouble. Also, cheating just makes stuff worse for you.
622.1 ish
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.
Verified answer:622
To solve this problem, let's first understand the given information. It states that we have an equilateral triangle with a side length of 14 centimeters. An equilateral triangle is a triangle with all three sides equal in length.
We are asked to find the volume of the cone formed by revolving this equilateral triangle about one of its altitudes. To visualize this, imagine taking the equilateral triangle and rotating it around one of its sides. This rotation will create a three-dimensional cone.
Now, let's calculate the volume of the cone. The formula for the volume of a cone is given by:
V = (1/3) * π * r^2 * h
Where:
V = Volume of the cone
π = Pi (approximately 3.14)
r = Radius of the base of the cone
h = Height of the cone
In our case, the radius of the base of the cone is half the length of the side of the equilateral triangle (14 centimeters), divided by 2. So, r = (14 cm) / 2 = 7 centimeters.
The height of the cone can be found using the Pythagorean theorem. By drawing an altitude in the equilateral triangle, we create a right-angled triangle. The height of the cone is the length of this altitude.
In an equilateral triangle, the altitude bisects the base, forming two congruent right-angled triangles. The altitude acts as the height of one of these right-angled triangles, and the base acts as the hypotenuse.
Using the Pythagorean theorem, we can find the height of one right-angled triangle (which is the height of the cone) as follows:
(Height)^2 + (Base/2)^2 = Side^2
(Height)^2 + (7 cm)^2 = (14 cm)^2
(Height)^2 = (14 cm)^2 - (7 cm)^2
(Height)^2 = 196 cm^2 - 49 cm^2
(Height)^2 = 147 cm^2
Taking the square root of both sides, we find:
Height = √(147 cm^2)
Height ≈ 12.1 cm
Now that we have the radius (r = 7 cm) and the height (h ≈ 12.1 cm), we can calculate the volume of the cone:
V = (1/3) * π * (7 cm)^2 * 12.1 cm
V ≈ (1/3) * 3.14 * (49 cm^2) * 12.1 cm
V ≈ 636 cm^3
Therefore, the number of cubic centimeters in the volume of the cone is approximately 636 (rounded to the nearest whole number).
is this for artofproblemsolving? I have the same question. Idk but I think it is close to 343√3/3π.
Volume of any right cone is 1/3 base area*height
volume=PI*7^2*(7*sqrt3)