A right circular cone has a constant slant height of 12 feet. Find the radius of the base for which the cone has maximum value
To find the radius of the base for which the cone has maximum value, we need to set up an equation relating the dimensions of the cone. Let's denote the radius of the base as r and the height of the cone as h.
Given that the slant height (l) of the cone is constant at 12 feet, we can use the Pythagorean theorem in the right triangle formed by the slant height, height, and radius.
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (slant height) is equal to the sum of the squares of the other two sides:
l^2 = r^2 + h^2
Since we know the slant height is constant at 12 feet, we can substitute l with 12:
12^2 = r^2 + h^2
144 = r^2 + h^2
Now, we need to express the height (h) in terms of the radius (r) to eliminate one variable from the equation. We can use similar triangles to find this relationship.
In a right circular cone, the slant height (l), height (h), and radius (r) form a right triangle. The ratio of the height to the slant height is the same as the ratio of the radius to the slant height, since the triangles formed by the heights and radii are similar.
Thus, we have the following proportion:
h/l = r/l
Cross-multiplying, we get:
h = rl/l
Simplifying, we have:
h = r
Now, we can substitute h = r into the equation 144 = r^2 + h^2:
144 = r^2 + r^2
144 = 2r^2
Dividing both sides by 2:
72 = r^2
Taking the square root of both sides:
r = √72
Simplifying the radical:
r = √(36 * 2)
r = √36 * √2
r = 6√2
Therefore, the radius of the base for which the cone has the maximum value is 6√2 feet.