An open bucket is in the form of a frustum of a cone,whose radii of bottom and top are 7 and 28cm.If the capacity of the bucket is 21560cmcube,then find the cost of metal sheet used in making the bucket at the rate of Rs. 1per10cmsq.
the area of metal is the area of the small circle plus the area of the sides.
The small circle has area 49π
Now for the sides. Consider the entire cone with base radius 28 and height 4h.
Chop off the top 1/4 of the height, and the missing volume is (1/3)π*7^2*h
The entire cone would have volume (1/3)π*28^2*(4h)
So, calculate the height of the bucket (3h), knowing that
π/3 (28^2*4h-7^2*h) = 21560
Now, use that height to figure the area of metal:
π*28*√(28^2+(4h)^2)-π*7√(7^2+h^2) + π*7^2
To find the cost of the metal sheet used in making the bucket, we need to calculate the total surface area of the frustum of the cone.
The formula for the lateral surface area of a frustum of a cone is given by:
A = π(R1 + R2)l
Where R1 and R2 are the radii of the bottom and top of the frustum, and l is the slant height of the frustum.
To find the slant height of the frustum, we can use the Pythagorean theorem:
l^2 = (h^2) + (R2 - R1)^2
Where h is the height of the frustum, and R2 - R1 is the difference in radii.
The formula for the volume of a frustum of a cone is given by:
V = (1/3)πh(R1^2 + R2^2 + R1 * R2)
Given that the capacity of the bucket is 21560 cm^3, we can substitute this value for V in the volume formula and solve for h.
21560 = (1/3)πh(7^2 + 28^2 + 7 * 28)
Simplifying the equation, we get:
21560 = (1/3)πh(49 + 784 + 196)
21560 = (1/3)πh(1029)
21560 = (343/3)πh
Dividing both sides by (343/3)π, we get:
h = 21560 / ((343/3)π)
Now that we have the value of h, we can calculate the slant height l using the Pythagorean theorem equation mentioned earlier.
Once we have the values of R1, R2, and l, we can calculate the lateral surface area A using the formula A = π(R1 + R2)l.
Finally, to find the cost of the metal sheet, we divide the total surface area by 10 to get the cost in square centimeters, and then multiply by the rate of Rs. 1 per 10 cm^2.