A cylinder is such that the sum of its height and circumference of its base is 10 metres. Find the maximum volume of the cylinder.

Answer

To find the maximum volume of a cylinder given that the sum of its height and the circumference of its base is 10 meters, we can follow these steps:

### Step 1: Set up the relationship
We know that the circumference \( C \) of the base of the cylinder is given by:
\[ C = 2\pi r \]
where \( r \) is the radius of the base. According to the problem, the sum of the height \( h \) and the circumference is 10 meters:
\[ h + 2\pi r = 10 \]
From this equation, we can express the height \( h \) in terms of \( r \):
\[ h = 10 - 2\pi r \]

### Step 2: Write the volume formula
The volume \( V \) of a cylinder is given by:
\[ V = \pi r^2 h \]
Substituting the expression for \( h \) from Step 1 into the volume formula gives:
\[ V = \pi r^2 (10 - 2\pi r) \]
Expanding this, we have:
\[ V = 10\pi r^2 - 2\pi^2 r^3 \]

### Step 3: Differentiate the volume with respect to \( r \)
To find the maximum volume, we need to differentiate \( V \) with respect to \( r \) and set the derivative equal to zero:
\[ \frac{dV}{dr} = 20\pi r - 6\pi^2 r^2 \]
Setting the derivative equal to zero:
\[ 20\pi r - 6\pi^2 r^2 = 0 \]
Factoring out \( 2\pi r \):
\[ 2\pi r (10 - 3\pi r) = 0 \]
This gives us two solutions:
1. \( r = 0 \) (not valid for a cylinder)
2. \( 10 - 3\pi r = 0 \) which leads to:
\[ r = \frac{10}{3\pi} \]

### Step 4: Determine the maximum volume
To confirm that this value of \( r \) gives a maximum, we can check the second derivative:
\[ \frac{d^2V}{dr^2} = 20\pi - 12\pi^2 r \]
Substituting \( r = \frac{10}{3\pi} \):
\[ \frac{d^2V}{dr^2} = 20\pi - 12\pi^2 \left(\frac{10}{3\pi}\right) = 20\pi - 40\pi = -20\pi \]
Since this is negative, it confirms that we have a maximum.

### Step 5: Calculate the maximum volume
Now we can substitute \( r = \frac{10}{3\pi} \) back into the volume formula:
\[ h = 10 - 2\pi \left(\frac{10}{3\pi}\right) = 10 - \frac{20}{3} = \frac{10}{3} \]
Now substituting \( r \) and \( h \) into the volume formula:
\[ V = \pi \left(\frac{10}{3\pi}\right)^2 \left(\frac{10}{3}\right) \]
Calculating this:
\[ V = \pi \cdot \frac{100}{9\pi^2} \cdot \frac{10}{3} = \frac{1000}{27\pi} \text{ cubic meters} \]

### Final Answer
The maximum volume of the cylinder is:
\[ \boxed{\frac{1000}{27\pi}} \text{ cubic meters} \]