A container in the shape of a right circular cylinder with no top has surface area 3π
square feet. What height h and base radius r will maximize the volume of the cylinder, and what is that volume?
a) The maximum volume occurs when the container has base radius 1 foot and height 2 feet. The maximum volume of the container is 2π3 cubic feet.
b) The maximum volume occurs when the container has base radius 1 foot and height 2 feet. The maximum volume of the container is 2π
cubic feet.
c)The maximum volume occurs when the container has base radius 1 foot and height 1 foot. The maximum volume of the container is π3 cubic feet.
d)The maximum volume occurs when the container has base radius 1 foot and height 1 foot. The maximum volume of the container is π
cubic fee
To solve this problem, we need to use the formulas for the surface area and volume of a right circular cylinder.
The surface area is given by the formula:
A = 2πrh + πr^2
We're given that the surface area is 3π. So we can set up the equation:
3π = 2πrh + πr^2
We want to maximize the volume, V, which is given by the formula:
V = πr^2h
To find the values of r and h that maximize the volume, we need to first solve the equation for r in terms of h, and then substitute that value into the volume formula.
From the surface area equation, we can rearrange it to solve for r:
3π = 2πrh + πr^2
3 = 2h + r
r = 3 - 2h
Now we can substitute this value of r into the volume formula:
V = π(3 - 2h)^2h
V = π(9 - 12h + 4h^2)h
V = π(4h^3 - 12h^2 + 9h)
To find the maximum volume, we need to find the critical points of this volume function. We can take the derivative of V with respect to h and set it equal to zero:
dV/dh = 0
4π(12h^2 - 24h + 9) = 0
Simplifying the equation, we get:
12h^2 - 24h + 9 = 0
This quadratic equation can be factored as:
(3h - 1)(4h - 9) = 0
Setting each factor equal to zero, we get:
3h - 1 = 0 OR 4h - 9 = 0
Solving each equation for h, we find two possible values:
h = 1/3 OR h = 9/4
Since the height cannot be negative, we can ignore the solution h = 9/4.
So the height that maximizes the volume is h = 1/3.
Substituting this value of h into the equation for r:
r = 3 - 2(1/3)
r = 3 - 2/3
r = 7/3
Therefore, the base radius that maximizes the volume is r = 7/3 and the height is h = 1/3.
To find the maximum volume, we can substitute these values into the volume formula:
V = π(4(1/3)^3 - 12(1/3)^2 + 9(1/3))
Simplifying the expression, we get:
V = π(4/27 - 4/9 + 3)
V = π(4/27 - 12/27 + 3)
V = π(-8/27 + 3)
V = π(17/27)
Therefore, the maximum volume of the container is 17π/27 cubic feet.
The correct answer is:
c) The maximum volume occurs when the container has base radius 7/3 feet and height 1/3 feet. The maximum volume of the container is 17π/27 cubic feet.