Final answer:
The maximum possible volume of a right circular cylinder with a total surface area of 150π can be found by maximizing the cubic function.
By taking the derivative of the volume formula with respect to r and solving for r, we can determine the value of r that yields the maximum volume.
Plugging this value of r back into the volume formula gives us the maximum possible volume: 250π cubic units.
Step-by-step explanation:
In this case, the total surface area is given as 150π.
We can substitute this value into the formula and solve for r or h by rearranging the equation.
Let's solve it for r:
150π = 2πr² + 2πrh
Divide both sides by 2π:
75 = r² + rh
We can express h in terms of r by rearranging the equation:
h = (75 - r²)/r
Now, substitute the value of h back into the volume formula:
V = πr²h
V = πr²[(75 - r²)/r]
Simplify the expression:
V = 75πr - πr³
To find the maximum volume, we need to find the maximum value of this cubic function.
To do this, we can take the derivative of V with respect to r and set it equal to zero:
dV/dr = 75π - 3πr²
= 0
Solve for r:
r² = 25
r = ±5
Since we are looking for the maximum volume, we take the positive value of r:
r = 5
Now, substitute this value of r back into the equation for h:
h = (75 - 25)/5
h = 10
Therefore, the maximum possible volume of the right circular cylinder is given by:
V = π(5)²(10)
= 250π cubic units