The area of the canvas cloth needed to erect a right conical tent of height 20ft. and circular base of circumference 30π ft. is:

The correct option is 375π sq.ft.

To find the area of the canvas cloth required to erect a right conical tent, we need to calculate the curved surface area (CSA) of the cone. The curved surface area is given by the formula:
$\text{CSA}=\pi rl$
where $r$ is the radius of the circular base and $l$ is the slant height of the cone.

Step 1: Find the radius of the circular base
The circumference of the circular base is given as 30π ft. The formula for the circumference of a circle is:
$C=2\pi r$
Equating the two values:
$2\pi r=30\pi$
Dividing both sides by $2\pi$:
$r=15\text{ft.}$

Step 2: Find the slant height of the conical tent
The height of the conical tent ($h$) is given as 20 ft. The relationship between radius ($r$), height ($h$), and slant height ($l$) of a right circular cone is given by Pythagoras' theorem:
$l=\sqrt{r^2+h^2}$
Substituting the values of $r$ and $h$:
$l=\sqrt{{15}^2+{20}^2}$
$l=\sqrt{225+400}$
$l=\sqrt{625}$
$l=25\text{ft.}$

Step 3: Calculate the area of the canvas cloth (CSA)
Using the formula for the curved surface area:
$\text{CSA}=\pi \times r\times l$
Substituting $r=15$ ft. and $l=25$ ft.:
$\text{CSA}=\pi \times 15\times 25$
$\text{CSA}=375\pi \text{sq.ft.}$

Thus, the area of the canvas cloth needed to erect the conical tent is 375π sq.ft.