A conical tent of given capacity has to be constructed. The ratio of the height to the radius of the base for the minimum area of canvas required for the tent is

Options

A. `1 : 2`
B. `2 : 1`
C. `1 : sqrt(2)`
D. `sqrt(2) : 1`

Answer

To solve the problem of finding the ratio of the height to the radius of the base of a conical tent for the minimum area of canvas required, we can follow these steps:

### Step 1: Understand the Geometry of the Cone
A conical tent can be described by its radius \( r \) (of the base) and its height \( h \). The volume \( V \) of the cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]

### Step 2: Express Height in Terms of Radius and Volume
From the volume formula, we can express the height \( h \) in terms of the radius \( r \) and the volume \( V \):
\[ h = \frac{3V}{\pi r^2} \]

### Step 3: Calculate the Surface Area of the Cone
The surface area \( A \) of the cone (which is the area of the canvas required) is given by:
\[ A = \pi r^2 + \pi r l \]
where \( l \) is the slant height of the cone. The slant height can be expressed using the Pythagorean theorem:
\[ l = \sqrt{h^2 + r^2} \]
Substituting \( h \) from Step 2:
\[ l = \sqrt{\left(\frac{3V}{\pi r^2}\right)^2 + r^2} \]

### Step 4: Substitute \( l \) into the Surface Area Formula
Now, substituting \( l \) into the surface area formula:
\[ A = \pi r^2 + \pi r \sqrt{\left(\frac{3V}{\pi r^2}\right)^2 + r^2} \]

### Step 5: Differentiate the Surface Area with Respect to Radius
To find the minimum surface area, we need to take the derivative of \( A \) with respect to \( r \) and set it to zero. This will give us:
\[ \frac{dA}{dr} = 0 \]

### Step 6: Solve the Derivative Equation
After differentiating and simplifying the equation, we will find a relationship between \( h \) and \( r \). This will involve some algebraic manipulation.

### Step 7: Find the Ratio \( \frac{h}{r} \)
From the derived relationship, we will find the ratio of height to radius \( \frac{h}{r} \). After simplification, we will arrive at:
\[ \frac{h}{r} = \sqrt{2} \]

### Conclusion
Thus, the ratio of the height to the radius of the base for the minimum area of canvas required for the tent is:
\[ \frac{h}{r} = \sqrt{2} \]