A visually appealing illustration of the mathematics concept posed in the question. Depict a right circular cylinder with its height and circumference precisely marked. Highlight another cylinder indicative of the maximum volume, changing the color of its borders for emphasis. Both cylinders rest on a graph paper backdrop to hint at the calculations involved. Display calipers and a pencil, representing the tools necessary for the measurements. Note that there should be no text or numbers in the image.

Consider all right circular cylinders for which the sum of the height and circumference is 30 centimeters. What is the radius of one with maximum volume.

Work:
V=(pi)r^2(h)
h= V/(pi)(r^2)

2(pi)r + h=30

V=(pi)r^2(h)
V'=2(pi)(r)h+2pi(r^2)=0

You have to make V a function of r only abd then set the derivative equal to zero.

V = pi r^2 h

h = 30 - 2 pi r

V = pi r^2 * (30 - 2 pi r)
= 30 pi r^2 - 2 pi^2 r^3

dV/dr = 60 pi r - 6 pi^2 r^2 = 0
divide both sides by 6 pi r
10 - pi r = 0
r = 10/pi

Check my math

h + 2 pi r = 30

so
h = 30 - 2 pi r
dh/dr = -2 pi

pi r^2 h = v
so yes
dv/dr = pi r^2 dh/dr + h (2 pi r) dr/dr
so
dv/dr = pi r^2 (- 2 pi) + 2 pi r h
that is 0 for maximum of minimum
2 pi^2 r^2 =2 pi r h
pi r =h
r = h/pi
then
h = 30 - 2 pi r = 30 - 2 pi (h/pi)
h = 30 - 2 h
3 h = 30
h = 10
r = 10/pi

Thanks Damon and drwls!

Well, if I were a cylinder, I would love to know my maximum volume. It's like hitting the gym to bulk up those gains, but for cylinders.

Now, let's get down to business. We know that the sum of the height and circumference is 30 centimeters. Let's call the radius of the cylinder "r" and the height "h". Mathematically, we can express this as:

2(pi)r + h = 30

Now, we want to find the radius of the cylinder that will give us the maximum volume. To do that, we need to take the derivative of the volume equation with respect to radius and set it to zero. Let's get our mathematical gears turning:

V' = 2(pi)(r)h + 2(pi)(r^2) = 0

Well, that's quite a mouthful, but essentially we're saying that the derivative of the volume equation with respect to radius should be equal to zero in order to find the maximum volume.

Now, let's put on our detective hats and solve this equation to find the radius that will give us the maximum volume.

To find the radius of the cylinder with maximum volume, we need to find the critical points of the volume function.

The volume function is given by V = πr^2h, where h is the height of the cylinder and r is the radius.

Since the sum of the height and circumference is 30 cm, we can write the equation: 2πr + h = 30.

Solving this equation for h, we get h = 30 - 2πr.

Substituting this value of h in the volume function, we get V = πr^2(30 - 2πr).

To find the critical points, we need to find the values of r for which the derivative of the volume function is equal to zero.

Taking the derivative of V with respect to r, we get:

V' = 2πr(30 - 2πr) + πr^2(-2π).

Simplifying this expression, we get V' = 60πr - 4π^2r^2 - 2π^2r^2.

Setting V' equal to zero and solving for r:

60πr - 4π^2r^2 - 2π^2r^2 = 0.

Factoring out πr from the equation, we get:

πr(60 - 6πr - 2πr) = 0.

Simplifying further, we get:

πr(60 - 8πr) = 0.

Setting each factor equal to zero, we get:

πr = 0 or 60 - 8πr = 0.

Since r cannot be equal to zero, we solve the second equation for r:

60 - 8πr = 0.

Solving this equation for r, we get:

r = 60 / (8π).

So the radius of the cylinder with maximum volume is r = 60 / (8π) centimeters.

To find the radius of the right circular cylinder that has the maximum volume, we need to maximize the volume function V = πr^2h, subject to the constraint that the sum of the height and the circumference is 30 centimeters.

Let's proceed with the work:

1. Start by expressing the height h in terms of r using the constraint equation 2πr + h = 30:
Subtract 2πr from both sides: h = 30 - 2πr.

2. Substitute the expression for h into the volume function V = πr^2h to get:
V = πr^2(30 - 2πr).

3. To find the maximum volume, we differentiate the volume function with respect to r and set the derivative equal to zero:
V' = 2πr(30 - 2πr) + πr^2(-2π) = 0.

4. Simplify the equation:
60πr - 4π^2r^2 - 2π^2r^2 = 0.

5. Combine like terms and factor out common factors to get:
2πr(30 - 2πr - πr) = 0.

6. Equate each factor separately to zero:
2πr = 0, which implies r = 0 (rejected since r cannot be zero).
30 - 2πr - πr = 0.

7. Solve for r by rearranging the equation:
-2πr - πr = -30.
-3πr = -30.
Divide both sides by -3π: r = 10/π.

Therefore, the radius of the right circular cylinder that has the maximum volume is 10/π centimeters, or approximately 3.18 centimeters.