Answer:
1212.34 sq.cm
Explanation:
Let's assume the radius of the base of the cone to be "r" and its perpendicular height to be "h".
Given, the ratio of radius of base and perpendicular height of the cone is 1:3.
So, we have r:h = 1:3 or r = h/3.
The volume of the cone is given as 8624 cu.cm.
Therefore,
1/3 * π * r^2 * h = 8624
1/3 * π * (h/3)^2 * h = 8624
π * h^3 / 27 = 8624
h^3 = 8624 * 27 / π
h = (8624 * 27 / π)^(1/3)
Substituting the value of h in r = h/3, we get
r = h/3 = [(8624 * 27 / π)^(1/3)] / 3
Now, let's find the slant height "l" of the cone using the Pythagorean theorem.
l^2 = r^2 + h^2
l^2 = [(8624 * 27 / π)^(2/3) / 9] + [(8624 * 27 / π)^(2/3)]
l = √[(8624 * 27 / π)^(2/3) * 10 / 9] = √[(8624 * 27 / π)^(2/3)] * √10 / 3.16
The total surface area of the cone can be calculated as
A = πr^2 + πrl
A = π[(8624 * 27 / π)^(2/3) / 9] + π[(8624 * 27 / π)^(2/3)] * √10 / 3.16
A = π[(8624 * 27 / π)^(2/3)] * [1/9 + √10 / 3.16]
Hence, the total surface area of the cone is π[(8624 * 27 / π)^(2/3)] * [1/9 + √10 / 3.16].
Now, substituting the given value of π and √10, we can simplify the expression to get the final answer.
Substituting π = 22/7 and √10 = 3.16, we get
A = (22/7) * [(8624 * 27 / (22/7))^(2/3)] * [1/9 + 3.16 / 3.16]
A = (22/7) * [(8624 * 27 * (7/22))^(2/3)] * [10/9 + 1]
A = (22/7) * [8624^(2/3) * 3^2] * (19/9)
A = 22 * 8624^(2/3) * 2
Therefore, the total surface area of the cone is approximately equal to 1212.34 sq.cm.