A cylindrical jar without a lid has to be constructed using a given surface area of a metal sheet It the capacity of the jar is to maximum, then the diameter of the jar must be k times the height of the jar The value of k is:

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem of maximizing the capacity of a cylindrical jar without a lid, we will follow these steps:

### Step 1: Understand the Surface Area and Volume Formulas
For a cylindrical jar without a lid, the surface area \( S \) and volume \( V \) can be expressed as:
- Surface Area: \( S = 2\pi rh + \pi r^2 \) (where \( r \) is the radius and \( h \) is the height)
- Volume: \( V = \pi r^2 h \)

### Step 2: Relate Diameter and Height
According to the problem, the diameter \( D \) of the jar is \( k \) times the height \( h \). Since the diameter is twice the radius, we have:
\[ D = 2r = k \cdot h \]
From this, we can express the radius in terms of height:
\[ r = \frac{kh}{2} \]

### Step 3: Substitute for Height in the Surface Area Formula
We can substitute \( r \) in the surface area formula. First, we need to express \( h \) in terms of \( r \):
From the surface area formula:
\[ S = 2\pi rh + \pi r^2 \]
Rearranging gives us:
\[ h = \frac{S - \pi r^2}{2\pi r} \]

### Step 4: Substitute \( h \) in the Volume Formula
Now, we substitute \( h \) back into the volume formula:
\[ V = \pi r^2 \left( \frac{S - \pi r^2}{2\pi r} \right) \]
This simplifies to:
\[ V = \frac{r(S - \pi r^2)}{2} \]

### Step 5: Differentiate the Volume with Respect to \( r \)
To find the maximum volume, we differentiate \( V \) with respect to \( r \):
\[ \frac{dV}{dr} = \frac{d}{dr} \left( \frac{Sr}{2} - \frac{\pi r^3}{2} \right) \]
This gives us:
\[ \frac{dV}{dr} = \frac{S}{2} - \frac{3\pi r^2}{2} \]

### Step 6: Set the Derivative Equal to Zero
To find the critical points, set the derivative equal to zero:
\[ \frac{S}{2} - \frac{3\pi r^2}{2} = 0 \]
This leads to:
\[ S = 3\pi r^2 \]

### Step 7: Substitute Back to Find \( h \)
Now, substitute \( S \) back into the equation for \( h \):
Using \( S = 3\pi r^2 \):
\[ h = \frac{3\pi r^2}{2\pi r} - \frac{r}{2} \]
This simplifies to:
\[ h = \frac{3r}{2} - \frac{r}{2} = r \]

### Step 8: Relate Diameter to Height
Since \( h = r \), we can now express the diameter:
\[ D = 2r = 2h \]
Thus, we can conclude:
\[ k = 2 \]

### Final Answer
The value of \( k \) is \( 2 \).

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