a bucket i sin the form of frustum with the capacity of 12308.8 cm cube.the radii of the top and bottom of circular ends are 12 cm and 20 cm resp.find the hieght of the bucket and also the area of metal sheets used in making the bucket.(use pie=3.14)
Volume of the frustum (V) = 12308.8 cm3
Also, radii of the top (r1) = 12 cm
Radii of the bottom (r2) = 20 cm
We know that
Volume of frustum,
Also, we know that
Now, the bucket will be open at the top and so the area of metal sheet used in making the bucket (Say A)
A = lateral surface of the frustum + (Area of circle at the bottom with r2 = 20 cm)
For two cones with base radii r1, r2 respectively and r1 < r2, let s1, s2 be the slant heights.
For the frustum which is the the difference between the two cones, let:
A be the surface area excluding that of the ends,
s be the slant height,
h be the vertical height.
From similar triangles:
s1 = s r1 / (r2 - r1) ...(1)
s2 = s r2 / (r2 - r1) ...(2)
A = pi(r2 s2 - r1 s1)
Substituting for s1 from (1) and s2 from (2):
A = pi s(r2^2 - r1^2) / (r2 - r1)
A = pi s(r1 + r2) ...(3)
Dropping a perpendicular from a point on the circumference of the smaller end of the frustum on to the larger end forms a right angled triangle in which:
s^2 = (r2 - r1)^2 + h^2
Eliminating s from (3):
A = pi(r1 + r2) sqrt[ (r2 - r1)^2 + h^2 ].
Adding the base area pi 12^2 of the bucket gives for the area of sheet metal:
pi 12^2 + pi(12 + 20) sqrt[ (20 - 12)^2 + 15^2 ]
= 2160 cm^2 to 3 sig. fig.