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 the maximum value, we need to use calculus to determine the maximum.
Let's denote the radius of the base as r, and the height of the cone as h. We are given that the slant height (l) is constant at 12 feet.
The slant height (l) of a cone is related to the radius (r) and height (h) using the Pythagorean theorem:
l^2 = r^2 + h^2
Since we are given that the slant height is constant at 12 feet, we can write:
144 = r^2 + h^2
Now, we need to express the volume of the cone (V) in terms of r. The volume of a right circular cone is given by:
V = (1/3) * π * r^2 * h
Since we know that h^2 = 144 - r^2 (from the Pythagorean theorem equation), we can substitute this into the volume equation to express the volume in terms of r:
V = (1/3) * π * r^2 * (144 - r^2)
To find the maximum value of V with respect to r, we need to find the critical points of the volume function. By taking the derivative of V with respect to r and setting it equal to zero:
dV/dr = 0
We can find the critical points.
Differentiating the volume function with respect to r, we get:
dV/dr = (1/3) * π * (2r(144 - r^2) - r^2 * 2r)
Simplifying this expression:
dV/dr = (1/3) * π * (288r - 3r^3 - 2r^3)
dV/dr = (1/3) * π * (288r - 5r^3)
Setting this derivative equal to zero and solving for r:
(1/3) * π * (288r - 5r^3) = 0
288r - 5r^3 = 0
Factoring out r:
r(288 - 5r^2) = 0
Setting each factor equal to zero:
r = 0 or 288 - 5r^2 = 0
From the second equation, we can solve for r^2:
288 - 5r^2 = 0
5r^2 = 288
r^2 = 57.6
Taking the positive square root:
r = √57.6 ≈ 7.6 feet
Therefore, the radius of the base for which the cone has the maximum value is approximately 7.6 feet.