A right circular cone has a fixed slant height, s= 12 inches. Determine the radius and the height of the cone with maximum volume
volume of cone is
V=4/3(pi)r^2h
s^2=r^2+h^2 or s^2-h^2=r^2
V=4/3(pi)(s^2-h^2)h
if 4/3(pi)=k
V=k(144-h^2)h
V=144kh-kh^3
volume is a max when dV/dh is zero
dV/dh = 144k-3kh^2
144k-3kh^2 = 0
144k = 3kh^2
48=h^2
this gives you h, from which you can find r from
s^2-h^2=r^2
please check my maths!
I am sure that Dr Russ meant to type
volume of cone is
V=1/3(pi)r^2h
Yes oops!
To determine the radius and height of the cone with maximum volume, we need to find the dimensions that optimize the volume function.
Note that the formula for the volume of a right circular cone is V = (1/3)πr^2h, where r is the radius and h is the height.
First, let's find an expression for the volume of the cone in terms of one variable, either r or h.
Using the Pythagorean theorem, we can relate the slant height (s) to the radius (r) and the height (h) of the cone. The relationship is represented by the equation s^2 = r^2 + h^2.
Since we have a fixed slant height (s = 12 inches), we can substitute this value into the equation and solve for one of the variables. Let's solve for h:
12^2 = r^2 + h^2
144 = r^2 + h^2
Now, let's solve for h in terms of r:
h^2 = 144 - r^2
h = √(144 - r^2)
Now, we substitute this expression for h into the volume formula:
V = (1/3)πr^2h
V = (1/3)πr^2√(144 - r^2)
Now, we have the volume function in terms of a single variable, r. To find the maximum volume, we can take the derivative of the volume function with respect to r, set it equal to zero, and solve for r.
dV/dr = (1/3)π(2r)√(144 - r^2) + (1/3)πr^2(-1/2)(144 - r^2)^(-1/2)(-2r)
dV/dr = (2/3)πr√(144 - r^2) - (2/3)πr^3 / √(144 - r^2)
Setting the derivative equal to zero:
(2/3)πr√(144 - r^2) - (2/3)πr^3 / √(144 - r^2) = 0
Simplifying:
2r√(144 - r^2) - r^3 / √(144 - r^2) = 0
2r^2 - r^3 / √(144 - r^2) = 0
Since we are looking for a local maximum, we must solve for r when the derivative is equal to zero and also check for any critical points in the interval 0 < r < 12.
Next, we need to solve the equation for r. However, finding the exact solution is quite challenging given the complexity of the equation. Therefore, we can use numerical methods or graphing technology to find approximate solutions.
Using a graphing calculator or software, graph the equation and find the x-coordinate of the local maximum point. This approximate value of r corresponds to the radius of the cone with maximum volume.
Once you have the value of r, substitute it back into the equation for h:
h = √(144 - r^2)
This will give you the approximate height of the cone with maximum volume.
Remember, this method provides an approximate solution because the exact solution involves solving a complicated equation.