What is the maximum possible volume of a right circular cylinder with a total surface area 600 Pi in.^2.?

Question

(including the top and the bottom)

Answer 1

Recall that the surface area of a cylinder is given by the following formula:

S = (area of the top and bottom) + (area of the sides).

Note that the surface area of the top and bottom is just plainly the area of a circle, (pi)r^2. We have 2 of them, so it's 2(pi)r^2.
The area of the sides is given to be the circumference of the ends, times height. OR, 2(pi)rh. Therefore,

S = 2pi(r^2) + 2pi(r)(h)

We can clean this up a bit.

S = 2pi [r^2 + rh]

We're given the surface area S; it's equal to 600pi. Therefore,

600pi = 2pi [r^2 + rh]

What we want to maximize is the volume of the cylinder. The volume of a cylinder is given by the following formula:

V = pi(r^2)(h)

However, we can express h in terms of r, from our surface area equation.

600pi = 2pi [r^2 + rh]. Divide both sides by 2pi, to get
300 = r^2 + rh. Subtract r^2 both sides,
300 - r^2 = rh. Divide both sides by r, to get
(300 - r^2)/r = h

Now, we replace what we got for h in the Volume formula.
V = pi(r^2)(h)
V = pi(r^2) [(300 - r^2)/r].

Notice that the r on the bottom will cancel, leaving us with

V = pi(r)[300 - r^2]. Expanding this, we get

V = 300pi(r) - pi(r^3).

Since we now have V expressed in terms of one variable, this will be our function, V(r).

V(r) = 300pi(r) - pi(r^3).

In order to maximize the volume, we have to calculate V'(r), and then make it 0.

V'(r) = 300pi - pi (3r^2)

Now, we make V'(r) = 0.

0 = 300pi - pi (3r^2)

And solve normally.

pi (3r^2) = 300pi. Divide both sides by 3pi, we get

r^2 = 100. Solving this, we get
r = {-10, 10}

However, the radius cannot be a negative value, so we discard the negative result, and

r = 10.

At this point we found WHERE the maximum volume occurs (which is when the radius is 10). What the question is asking for is WHAT the maximum volume is. The answer is as simple as plugging in r = 10 for V(r).

V(r) = 300pi(r) - pi(r^3)
V(10) = 300pi(10) - pi(10^3)
V(10) = 3000pi - 1000pi
V(10) = 2000pi

Therefore, the maximum possible volume of a right circular cylinder with a total surface area of 600 pi inches squared is
2000pi inches cubed.

Answer 2

Right Cylinder Surface Area

Answer 3

Given a right circular cylinder with

V = Volume
S = Surface Area
r = radius
h = height

V = πr²h
S = 2πr² + 2πrh = 600π in³

Solve for h in terms of the other variables.

V = πr²h
h = V/πr²

Plug that value for h into the formula for S.

S = 2πr² + 2πrh = 2πr² + 2πr(V/πr²) = 2πr² + 2V/r

Differentiate to solve for r to minimize S.

dS/dr = 4πr - 2V/r² = 0
4πr = 2V/r²
2πr = V/r²
r³ = V/2π
r = (V/2π)^(⅓)

Take the second derivative to see if this is a maximum or minimum.

d²S/dr² = 4π + 4V/r³ > 0 as both terms are positive. So it is a minimum as we had hoped.

Now solve for h in terms of r. Plug into the formula for h which was derived above.

h = V/πr² = 2V/2πr² = (2/r²)(V/2π) = (2/r²)(r³) = 2r

Now solve for r by plugging into the formula for S.

S = 2πr² + 2πrh = 2πr² + 2πr(2r) = 2πr² + 4πr² = 6πr²
600π = 6πr²
100 = r²
r² = 100
r = 10
h = 2r = 20

Now solve for the volume of the cylinder.

V = πr²h = π(10)²(20) = π(100)(20)
V = 2000π = 6283.1853 in³

Answer 4

Write the equation for the exterior area of an actual cylinder, and write the equation for the quantity of an actual cylinder. you could connect them, and you know that the exterior area is unquestionably 384 pi. that could desire to be adequate.

Answer 5

Why. Are you doing homework or somthin?

Answer 6

12000