A bucket is in the shape of a frustum of a cone. Its diameter at the top and the bottom are
38 cm and 28 cm respectively. Its depth is 20cm.Determine the full capacity of the bucket in litres.
(b) The above container was filled with water and later transferred to a cylinder of height 92cm.
Calculate to 1 d.p the radius of the cylinder.
To find the full capacity of the frustum of a cone, we need to find its volume.
The formula for the volume of a frustum of a cone is:
V = (1/3) * π * h * (r1^2 + r2^2 + (r1 * r2))
where V is the volume, h is the depth, r1 is the radius of the top surface, and r2 is the radius of the bottom surface.
Given that the diameter at the top is 38 cm (which means the radius is 19 cm) and the diameter at the bottom is 28 cm (which means the radius is 14 cm), and the depth is 20 cm, we can substitute these values into the formula:
V = (1/3) * π * 20 * (19^2 + 14^2 + (19 * 14))
V ≈ 1/3 * 3.14159 * 20 * (361 + 196 + 266)
V ≈ 1/3 * 3.14159 * 20 * 823
V ≈ 17233.91556 cm^3
V ≈ 17,233.92 cm^3
To convert this volume to liters, we need to divide by 1000 (since there are 1000 cm^3 in 1 liter):
17,233.92 cm^3 ÷ 1000 = 17.23392 liters
Therefore, the full capacity of the bucket is approximately 17.2 liters.
Now, let's move on to part (b).
To find the radius of the cylinder, we need to use the volume of the frustum (17,233.92 cm^3) and the height of the cylinder (92 cm).
The formula for the volume of a cylinder is:
V = π * r^2 * h
where V is the volume, r is the radius, and h is the height.
We can rearrange the formula to solve for r:
r = sqrt(V / (π * h))
Plugging in the given values:
r = sqrt(17,233.92 / (3.14159 * 92))
r = sqrt(185.945817 / 287.97907051)
r = sqrt(0.645721782)
r ≈ 0.803 cm
Therefore, the radius of the cylinder is approximately 0.8 cm.