A container open at the top is a right circular cylinder having a surface (side and lower base) area of 108 square centimeters. What should the radius and altitude be in order to provide the largest possible volume.
so you have
πr^2 + 2πrh = 108
so h = (108-πr^2)/(2πr)
Now, the volume is
v = πr^2 h = πr^2 (108-πr^2)/(2π) = 54r^2 - π/2 r^4
so max vomume occurs when
dv/dr = 0
dv/dr = 108r - 2πr^3 = 2r(54-πr^2)
dv/dr=0 at r=√(54/π) = 3√(6/π)
That makes the height h = (108-πr^2)/(2πr) = 27/(πr)
To find the maximum volume of a right circular cylinder, we need to optimize its dimensions by maximizing the surface area.
Let's denote the radius of the cylinder as 'r' and the altitude (height) as 'h'.
The surface area of the cylinder can be given by the formula:
Surface Area = 2πrh + πr^2
Given that the surface area is 108 square centimeters, we can write the equation as:
2πrh + πr^2 = 108
Now, to find the maximum volume, we need to maximize the volume function.
The volume of a cylinder is given by the formula:
Volume = πr^2h
We want to maximize the volume, so we can rewrite the equation as:
Volume = πr^2 (108 - 2h) [Substituting the value of surface area]
Now, let's differentiate the volume function with respect to the height 'h' to find the critical points:
dV/dh = 0
Differentiating, we get:
dV/dh = πr^2 (-2) = 0
From the above equation, we can see that dV/dh will be 0 when either r is 0 (which is not possible) or h is 0 (which is also not possible since the container must have some height for volume to exist).
Therefore, there are no critical points for maximizing the volume. This means that the maximum volume occurs when the derivative is either undefined or 0.
To find the maximum volume, we need to consider the endpoints of the feasible range.
Given that the container is open at the top, the altitude (height) is not restricted. It can be any positive value.
Hence, there is no upper limit for h. However, both r and h must be positive.
Now, to maximize the volume, we can consider the endpoint scenario where h = 0 because it is the lower bound for height.
When h = 0, the volume becomes 0.
Therefore, it is not possible to maximize the volume of the cylinder with height 0.
In conclusion, there is no maximum volume for a cylinder with a surface area of 108 square centimeters in the given scenario.
To find the dimensions of the right circular cylinder that will provide the largest possible volume, we need to optimize the volume equation.
Let's denote the radius of the cylinder as "r" and the height (or altitude) as "h".
The surface area of a right circular cylinder is given by the formula:
Surface Area = 2πrh + πr²
We are given that the surface area is 108 square centimeters, so we can write the equation:
2πrh + πr² = 108
Now, we want to find the dimensions that will maximize the volume of the cylinder. The volume of a right circular cylinder is given by the formula:
Volume = πr²h
To optimize the volume, we need to express it as a function of just one variable. Let's solve the surface area equation for h:
2πrh + πr² = 108
Divide both sides by πr:
2rh + r² = 108/π
Solve for h:
h = (108/π - r²)/(2r)
Now, substitute this expression for h in the volume equation:
Volume = πr²[(108/π - r²)/(2r)]
Simplify:
Volume = (1/2)r(108 - πr²)
To find the dimensions that will give the largest possible volume, we need to maximize this volume function.
We can do this by finding the critical points of the volume function, which are the values of r where the derivative of the volume function is equal to zero.
Differentiate the volume function with respect to r:
dV/dr = (1/2)(108 - πr²) - (1/2)r(2πr)
Set the derivative equal to zero:
(1/2)(108 - πr²) - (1/2)r(2πr) = 0
Simplify:
54 - (π/2)r² - πr² = 0
Combine like terms:
(54 - π)r² = 0
Now we have two possibilities:
1. (54 - π) = 0, which implies r² = 0. This corresponds to a degenerate cylinder with zero radius, which is not physically meaningful.
2. r² = 54/π
Since r must be positive in this context, we can ignore the negative root.
Take the square root of both sides:
r = sqrt(54/π) = (3√6)/√π
Now that we have the value of r, we can substitute it back into the surface area equation to find the corresponding value of h:
2πrh + πr² = 108
Substituting r = (3√6)/√π:
2π((3√6)/√π)h + π((3√6)/√π)² = 108
Simplifying:
6√6h + 3√6(3√6)/√π = 108
6√6h + 54/√π = 108
6√6h = 108 - 54/√π
h = (108 - 54/√π)/(6√6)
Therefore, the radius (r) should be (3√6)/√π and the altitude (h) should be (108 - 54/√π)/(6√6) in order to provide the largest possible volume of the right circular cylinder.