The total surface area of a cone whose radius is `(r)/(2)` and slant height 2l is

Options

A. `2 pir(l +r)`
B. `pir(l+(r)/(4))`
C. `pir(l+r)`
D. `2pirl`

Explanation

To find the total surface area of a cone with a radius of \( \frac{r}{2} \) and a slant height of \( 2l \), we can use the formula for the total surface area of a cone, which is given by:

\[ \text{Total Surface Area} = \pi r l + \pi r^2 \]

Where:
- \( r \) is the radius of the base of the cone,
- \( l \) is the slant height of the cone.

### Step 1: Identify the values
Here, we have:
- Radius \( r = \frac{r}{2} \)
- Slant height \( l = 2l \)

### Step 2: Substitute the values into the formula
Now, we can substitute these values into the total surface area formula:

\[ \text{Total Surface Area} = \pi \left(\frac{r}{2}\right) (2l) + \pi \left(\frac{r}{2}\right)^2 \]

### Step 3: Simplify the first term
Calculating the first term:

\[ \pi \left(\frac{r}{2}\right) (2l) = \pi \cdot \frac{r}{2} \cdot 2l = \pi r l \]

### Step 4: Simplify the second term
Now, calculating the second term:

\[ \pi \left(\frac{r}{2}\right)^2 = \pi \cdot \frac{r^2}{4} = \frac{\pi r^2}{4} \]

### Step 5: Combine both terms
Now, we can combine both terms to find the total surface area:

\[ \text{Total Surface Area} = \pi r l + \frac{\pi r^2}{4} \]

### Step 6: Factor out \( \pi \)
Factoring out \( \pi \):

\[ \text{Total Surface Area} = \pi \left( r l + \frac{r^2}{4} \right) \]

### Final Answer
Thus, the total surface area of the cone is:

\[ \text{Total Surface Area} = \pi \left( r l + \frac{r^2}{4} \right) \]