Question

(i) A 5-m-wide cloth is used to make a conical tent of base diameter 14 m and height 24 m. Find the cost of cloth used at the rate of Rs 25 per metre.

(ii) The radius and height of a solid right-circular cone are in the ratio of 5 : 12. If its volume is 314 $c{m}^3,$ find its total surface area. $[Take\pi =\mathrm{3.14.}]$

Answer 1

1-

Let the slant height of the cone be 'l' meters.
height = 24 m and diameter = 14 m or radius = 7 m.
Then according to the question.
l² = h² + r²
l² = 24² + 7²
l² = 576 + 49
l² = 625
l = √625
l = 25 m
So, the slant height of the cone is 25 m.
Now,
Curved Surface Area of the cone = πrl
=$\frac{22}{7}\times 7\times 25$
CSA = 550 $m^2$
Now, length of the cloth used = Area/ width of the cloth
= $\frac{550}{5}$
Length of the cloth used = 110 m
Cost of cloth = Rs. 25 per meter
Total cost of clothe used = 110$\times$25
= Rs. 2750
Answer.

2-Let proportionality constant =x
radius =5x and height =12x

now,
volume of cone =$\frac{\pi r^{2}h}{3}$
314=$\frac{3.14(5x{)}^2\times (12x)}{3}$
100=25 $\times 4$$x^3$
x=1
hence,
radius =5 cm
height =12 cm
so,
slant height=$\sqrt{(12{)}^2+(5{)}^2}$=13 cm
now,
total surface area =π$r^2$ + πrl
=(3.14 x 5 x 5) + 3.14 x 5 x 13 = (78.5 + 204.1)$c{m}^2$ = 282.6 $c{m}^2$