A closed cylindrical can has to be made with `100m^(2)` of plastic. If its volume is maximum, then the ratio of its radius to the height is

Options

A. `1:1`
B. `1:2`
C. `2:1`
D. `sqrt2:1`

Answer

To solve the problem of maximizing the volume of a closed cylindrical can made from 100 m² of plastic, we will follow these steps:

### Step 1: Define the variables
Let:
- \( r \) = radius of the cylinder
- \( h \) = height of the cylinder

### Step 2: Write the total surface area equation
The total surface area (TSA) of a closed cylinder is given by:
\[ TSA = 2\pi r h + 2\pi r^2 \]
Given that the TSA is 100 m², we have:
\[ 2\pi r h + 2\pi r^2 = 100 \]

### Step 3: Simplify the equation
We can simplify the equation by dividing everything by \( 2\pi \):
\[ rh + r^2 = \frac{100}{2\pi} \]
Let \( C = \frac{100}{2\pi} \), then:
\[ rh + r^2 = C \]

### Step 4: Express height in terms of radius
From the equation \( rh + r^2 = C \), we can express \( h \) in terms of \( r \):
\[ h = \frac{C - r^2}{r} \]

### Step 5: Write the volume equation
The volume \( V \) of the cylinder is given by:
\[ V = \pi r^2 h \]
Substituting \( h \) from the previous step:
\[ V = \pi r^2 \left(\frac{C - r^2}{r}\right) = \pi r(C - r^2) \]
This simplifies to:
\[ V = \pi (Cr - r^3) \]

### Step 6: Differentiate the volume with respect to radius
To find the maximum volume, we differentiate \( V \) with respect to \( r \):
\[ \frac{dV}{dr} = \pi \left(C - 3r^2\right) \]
Setting the derivative equal to zero for maximization:
\[ C - 3r^2 = 0 \]
Thus:
\[ 3r^2 = C \quad \Rightarrow \quad r^2 = \frac{C}{3} \]

### Step 7: Substitute back to find height
Now substitute \( C = \frac{100}{2\pi} \):
\[ r^2 = \frac{100}{6\pi} \quad \Rightarrow \quad r = \sqrt{\frac{100}{6\pi}} = \frac{10}{\sqrt{6\pi}} \]
Now substituting \( r \) back into the expression for \( h \):
\[ h = \frac{C - r^2}{r} = \frac{\frac{100}{2\pi} - \frac{100}{6\pi}}{r} \]
Calculating \( h \):
\[ h = \frac{\frac{300 - 100}{6\pi}}{r} = \frac{\frac{200}{6\pi}}{r} = \frac{100}{3\pi r} \]

### Step 8: Find the ratio of radius to height
Now, we find the ratio \( \frac{r}{h} \):
\[ \frac{r}{h} = \frac{r}{\frac{100}{3\pi r}} = \frac{3\pi r^2}{100} \]
Substituting \( r^2 = \frac{100}{6\pi} \):
\[ \frac{r}{h} = \frac{3\pi \left(\frac{100}{6\pi}\right)}{100} = \frac{3}{6} = \frac{1}{2} \]

### Final Answer
The ratio of the radius to the height is:
\[ \frac{r}{h} = \frac{1}{2} \]