There are 8436 steel balls, each with radius 1 centimeter, stacked in a tetrahedral pile, with one ball on top, 3 balls in the second layer, 6 in the third layer, 10 in the fourth, and so on. Determine the height of the pile in centimeters.

a)50√6/3
b)50√6/3 + 2
c)70√6/3
d)70√6/3 + 2
e)None of these

Two cones, such that one is of height H and radius R and other is of height H/2 and radius R/2, are joined with each other by vertex, forming a shape of an Hour-Glass. Initially Hour-Glass is kept such that the base of the smaller cone is facing upward in direction and is completely filled with sand. Rate of flow of sand from upper cone to lower cone is 1 cubic cm/sec. Hour-Glass is turned upside down, everytime when upper cone is emptied out. If the process is continued, what will be the height of sand after 8 minute 10 sec in lower cone?
Values are as given below:
R = 6cm
H = 30/π cm

(a)(10/π)*3^1/3
(b)(15/π)*5^1/3
(c)(20/π)*3^1/3
(d)(25/π)*5^1/3
(e)None of these

A tower has the following shape: a truncated right circular cone one with radii 2R (the lower base) and R ( the upper base) and height R bears a right circular cylinder whose radius is R, the height being 2R. Finally a semisphere of radius R is mounted on the cylinder. Suppose that the cross-sectional area S of the tower is given by f(x) where x is the distance of the cross-section from the lower base of the cone.
The range of f(x) is
a)[0,4πR²]
b)[πR²,4πR²]
c)[0,πR²]
d)[πR²,3πR²]

A tiled floor of a room has dimensions mxm sq. m. The dimensions of the tile used are nxn sq. m. All tiles used are green tiles except diagonal tiles which are red. After some time green tiles are replaced from 1 row and 1 column of tiles such that the red tiles are maximum. Find the number of green tiles after replacement.? (m not equal to n and the total number of tiles are odd)

a)(m²-4mn+2n²)/2n²
b)[(m-2n)² - 2n²]/n²
c)(m²-4mn-3n²)/n²
d)(m²-4mn+3n²)/n²

A fish tank is 100 cm long, 60 cm wide and 40 cm high.If it is tilted, resting on the 60 cm edge,then the water reaches the midpoint of the base.If it is then put down so that the base is horizontal again, what is the depth of the water?
a) 6cm
b) 8cm
c) 10cm
d) 12cm
e) Cannot be determined

Let A and B be two solid spheres such that the surface area of B is 300% higher than the surface area of A. The volume of A is found to be k% lower than the volume of B. The value of k must be:
a)85.5
b)92.5
c)90.5
d)87.5

From a circle of radius R an arc of 45^o is removed. Sector formed by this arc is folded such that both the straight edges of sector touch each other to form a cone .Find the volume of the cone
a) 7√7πR³/512
b) 5√7πR³/512
c) 3√7πR³/512
d) √7πR³/512

Problem: A cone of radius r and height 3r is cut from a solid cylinder of same radius and height.Now A hollow hemi-sphere of diameter r is dropped into the hollow created by removing the cone in such a way that the hollow part of the sphere is facing the base of the cone. Find the volume of water that can be filled in the hollow.
(a) 11/22 πr³
(b) 23/24 πr³
(c) 25/24 πr³
(d) 13/22 πr³