A container in the shape of a right circular cylinder with no top has surface area 3*pi (m2). What height h and base radius r will maximize the volume of the cylinder ?
surface area = πr^2 + 2πrh = 3π
r^2 + 2rh = 3
h = (3 - r^2)/(2r)
V = πr^2h = (πr/2)(3-r^2)
= (3π/2)r - (π/2)r^3
dV/dr = 3π/2 - (3/2)πr^2
= 0 for a max/min of V
(3π/2)r^2 = 3π/2
r^2 = 1
r = ± 1, but r > 0
r = 1
then h = (3-1)/2 = 1
check my arithmetic
To find the values of height (h) and base radius (r) that maximize the volume of the cylinder, we can use calculus and optimization techniques.
Let's denote the height of the cylinder as h and the base radius as r. The surface area of the cylinder is given as 3π m^2, which can be written as:
2πrh + πr^2 = 3π
Now, in order to maximize the volume, we need to express it in terms of a single variable. The volume of a cylinder is given by the formula:
V = πr^2h
We can rewrite the equation for surface area to solve for h:
2πrh = 3π - πr^2
h = (3 - r^2) / (2r)
Now, substitute this expression for h into the equation for volume:
V = πr^2 * ((3 - r^2) / (2r))
Simplifying, we have:
V = (3πr - πr^3/2)
To find the maximum volume, we need to find the critical points by taking the derivative of V with respect to r and setting it equal to zero:
dV/dr = 3π - (3/2)πr^2 = 0
Solving for r, we get:
3 - (3/2)r^2 = 0
r^2 = 2
r = √2
Now, substitute this value of r back into the equation for h:
h = (3 - (√2)^2) / (2√2)
h = 3 - 2 / (2√2)
h = (3 - √2) / √2
Therefore, to maximize the volume, the base radius should be √2 units and the height should be (3 - √2) / √2 units in order to achieve a surface area of 3π m^2.
To find the dimensions that maximize the volume of the cylinder given the surface area, we can set up equations based on the given information.
Let's assume the height of the cylinder is denoted as h and the base radius as r.
The formula for the surface area of a cylinder without the top is:
S = 2πrh
Given that the surface area is 3π, we have:
3π = 2πrh
Simplifying the equation, we get:
2rh = 3
Next, we need to find the formula for the volume of the cylinder. The formula for the volume of a cylinder is:
V = πr²h
Since we want to maximize the volume, we need to express it solely in terms of one variable. In this case, we can express h in terms of r using the equation we obtained earlier:
h = (3 / (2r))
Substituting this value into the equation for the volume, we have:
V = πr²(3 / (2r))
Simplifying further, we get:
V = (3πr²) / 2
Now, we have an equation for the volume of the cylinder solely in terms of r.
To maximize the volume, we can take the derivative of V with respect to r and set it equal to zero:
dV/dr = 0
(d/dx)[(3πr²) / 2] = 0
(6πr / 2) = 0
3πr = 0
r = 0
However, a cylinder with a radius of 0 does not exist, so we need to look for a maximum value within a specific range.
One way to do this is to consider the endpoints of the possible range for r. In this case, since the surface area is given as 3π square meters, the radius, r, must be positive and less than 1.
Therefore, the possible range for r is 0 < r < 1.
Next, we can evaluate the volume of the cylinder at the endpoints and compare them to the critical points we found earlier.
V(0) = (3π(0)²) / 2 = 0
V(1) = (3π(1)²) / 2 = (3π) / 2
Comparing the values, we find that V(1) is greater than V(0).
Therefore, the maximum volume occurs when r = 1.
Using the equation we derived earlier to express h in terms of r:
h = (3 / (2r))
we find:
h = (3 / (2(1)))
h = 3/2
So, the dimensions that maximize the volume of the cylinder are:
Height (h) = 3/2 meters
Base radius (r) = 1 meter