A tree trunk I metres long, is in the shape of a frustum of a cone, the radii of its ends being a and meters `(a > b).` It is required to cut from it a beam of uniforms quare section. Prove that the beam of the greatest volume that can be cut is `1/3 (al)/(a-b)` metres long.

A bucket of height 8 cm and made up of copper sheet is in the form of frustum of a right circular cone with radii of its lower and upper ends as 3 cm and 9 cm respectively. Calculate the height of the cone of which the bucket is a part. the volume of water which can be filled in the bucket. (iii) the area of copper sheet required to make the bucket.

A milk container is made of metal sheet in the shape of frustum of a cone whose volume is 10459(3)/(7)cm^(2). The radii of its lower and upper circular ends are 8cm and 20cm respectively. Find the cost of metal sheet used in making the container at the rate of Rs.1.40per square centimeter.

How many pieces of equal size can be cut from a rope of 30 metres long,each measuring 3(3)/(4) metres?

The vertical section of a road over a canal bridge in the direction of its length is in the form of circle of radius 8.9 metre. Then the greatest spped at which the car can cross this bridge wihout losing contact with the road at its hgihest point, the centre of gravity of the car being at a height h = 1.1 metre from the ground. Take g = 10 m/sec2-

A bead of mass m can move without friction along a long wire bent in a vertical plane in the shape of a graph of a certain function. Let l_(A) be the length of the segment of the wire from the origin to a certain point A. It is knonw that if the bead is released at point A such that l_(A) lt l_(A_(0)) , its motion will be strictly harmonic l (t) = l_(A) cos omega t . Prove that there exists a point B (l_(A_(0)) lt lt l_(B)) at which the condition of harmonicity of oscillations will be violated.

Along a yard 225 metres long,26 trees are planted at equal distances,one tree being at each end of the yard.What is the distance between two consecutive trees? (a) 8 metres (b) 9 metres (c)10 metres quad (d) 15 metres