$\text{What is the total surface area of a cone whose radius is}\frac{r}{4}\text{and slant height is}4l?$

The correct answer is $\pi r(1+\frac{r}{16})$.

Step-by-step explanation:

To find the total surface area of a cone, we calculate the sum of its curved surface area and its circular base area.

The general formula for the total surface area ($TSA$) of a cone with radius $R$ and slant height $L$ is given by:

$TSA=\pi RL+\pi R^2$

From the given problem, the dimensions of the cone are:

Radius of the base, $R=\frac{r}{4}$

Slant height, $L=4l$ (where $l=1$ unit)

Now, substitute these values into the total surface area formula:

$TSA=\pi ·\left(\frac{r}{4}\right)·\left(4\right)+\pi ·{\left(\frac{r}{4}\right)}^2$

Simplifying the curved surface area component:

$\text{Curved Surface Area}=\pi ·\frac{r}{4}·4=\pi r$

Simplifying the base area component:

$\text{Base Area}=\pi ·\frac{r^2}{16}$

Adding the two parts together gives:

$TSA=\pi r+\frac{\pi r^2}{16}$

Factoring out the common term $\pi r$:

$TSA=\pi r(1+\frac{r}{16})$

Hence, the total surface area of the cone is $\pi r(1+\frac{r}{16})$.