If the slant height of a right pyramid with square base is 4 metre and the total slant surface of the pyramid is 12 square metre, then the ratio of total slant surface and area of the base is :

Options

A. `16:3`
B. `24:5`
C. `32:9`
D. `12:3`

Answer

To solve the problem step by step, we need to find the ratio of the total slant surface area of the pyramid to the area of its base.

### Step 1: Understand the given values
We are given:
- Slant height (l) = 4 meters
- Total slant surface area (S) = 12 square meters

### Step 2: Formula for the total slant surface area of a pyramid
The formula for the total slant surface area of a right pyramid with a square base is given by:
\[ S = \frac{1}{2} \times \text{perimeter of base} \times \text{slant height} \]

For a square base, if the side length is \( a \), the perimeter (P) is:
\[ P = 4a \]

So, the formula for the slant surface area becomes:
\[ S = \frac{1}{2} \times 4a \times l \]
\[ S = 2a \times l \]

### Step 3: Substitute the known values
Now substituting the known values into the formula:
\[ 12 = 2a \times 4 \]
\[ 12 = 8a \]

### Step 4: Solve for the side length (a)
To find \( a \), we rearrange the equation:
\[ a = \frac{12}{8} = \frac{3}{2} \text{ meters} \]

### Step 5: Calculate the area of the base
The area (A) of the base of the pyramid, which is a square, is given by:
\[ A = a^2 = \left(\frac{3}{2}\right)^2 = \frac{9}{4} \text{ square meters} \]

### Step 6: Calculate the ratio of total slant surface area to area of the base
Now we can find the ratio of the total slant surface area to the area of the base:
\[ \text{Ratio} = \frac{S}{A} = \frac{12}{\frac{9}{4}} \]
To simplify this:
\[ \text{Ratio} = 12 \times \frac{4}{9} = \frac{48}{9} = \frac{16}{3} \]

### Final Answer
The ratio of the total slant surface area to the area of the base is:
\[ \frac{16}{3} \]

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