How many metres of cloth, 5 m wide will be required to make a conical tent, the radius of whose base is 7 m and height is 24 m?
Assuming that there is no wastage in cloth and no extra cloth is required at the seams the total area of cloth used will be equal to the lateral surface area (L) of the conical shaped tent. This is given by the formula:
L = (pi)*r*S
r = radius of the base of the tent = 7 m (given)
S = slant height
the slant height is given by the formula:
S = (r^2 + h^2)^(1/2)
where: h = height of tent = 24 m (given)
S = (7^2 +24^2)^(1/2) = (49 + 576)^(1/2) = 625^(1/2) = 25
Substituting the values of S and r in equation for slant area we get.
L = (22/7)*7*25 = 550 m^2
l = length of the cloth required to make the conical tent.
w = width of the cloth = 5 m given
L = w*l = 5*l
But as we have calculated above L = 550 m^2
Therefor: 550 = 5*l
Therefor: l = 550/5 = 110 m
Length of cloth required = 110 m
We assume the entire cloth is managed for the use of curved surface of the cone only and there is no wastage.
The area of the curved surface of the cone = pi*r*l = pi*r*sqrt(r^2+h^2), where l is the slant height and h is the vetical height of the cone, and r is the base radius.
r=7m and h = 24m are data for the conical tent
value of pi = 3.141592654 apprx.
Therefore, the the area of the curved surface of the conical tent = pi*7*sqrt(7^2+24^2) = 549.7787144 m^2.
Therefore, the cloth required to cover the curved surface of the conical ten = 549.7787144 m^2.
Therefore the required length of the cloth (with width 5m) = 549.7787144 m^2/(5m) = 109.9557429m