a well of radius r is dug 20m deep and the earth taken out is spead to form ambankment around it. the height of embankment is 5m and width is 1 m than find out r
Answers
option b
Explanation
the shape of well is cylinder
volume of earth dug out = πr^2h
= π r^2*20
= 20πr^2.........(1)
now volume of earth required for embankment
= base area x height
= π{r+1)^2 - r^2}*5
(since width of embankment is 1 m, radius of outer circle will be (r+1)m and base area of embankment will be area of outer circle minus area of inner circle)
= π(r^2 + 1 + 2r - r^2)*5
= π(2r+1)*5
= 5π(2r+1)........(2)
the volumes at (1) and (2) must be equal
20πr^2 = 5π(2r+1)
4r^2 = 2r + 1
4r^2 - 2r - 1 = 0
this is a quadratic equation where
a= 4, b = -2, c = -1
D = b^2 - 4ac
= (-2)^2 - 4*4*(-1)
= 4 + 16
D = 20
√D = √20 = 2√5
now
r = ( -b +_ √D)/2a
= (-(-2) +_ 2√5)/(2*4(
= (2 +_ 2√5)/8
= (1 +_√5)/4
now 1 - √5 is negative which is not a possible solution for radius of well. so we will take only positive solution
so, radius r = (1 + √5)/4
i.e. option b