The sum of the slant height of a cone and its height equals twice the difference of twice its height and its slant height.Find the ratio of its radius and height.
Answer: A Let the radius and the height of the cone be r and h respectively.Lets its alnt height be l l+h=2(difference of 2h and l) If 2h>l, l+h=2(2h-l) l=h which is not possible. If 2h< l, l+h=2(l-2h) 5h =l r=√(5h)2-h2 =2√6h r:h = 2√6:1
Q. No. 8:
A rectangle has a length of 60cm and a breadth of 40cm. If four squares each of side 4cm are cut from the four corners of the rectangle and the resulting figure is made into a cuboid, find the volume of the cuboid(in cubic cm).
Answer: B Dimensions of the cuboid are (60-2x)cm, (60-2x) and x cm volume of the cuboid = 2(2x)(30-x)(30-x) When the sum of two or more quantities is constant, their product will be maximum when the quantities are equal. The sum of 2x, 30-x and 30-x is constant. Their product is maximum when 2x=30-x x=10.
Q. No. 9:
In a cuboid, the sum of the squares of the three dimensions equals half its total surface area. Its volume is 729 cubic cm. Find its lateral surface area(in sq cm).
Answer: C Let the length, breadth and the height of the cuboid be lcm, bcm and hcm respectively. Given that l2+b2+h2 =1/2[lb+bh+hl] 1/2[(l-b)2+(b-h)2+(h-l)2]=0This is possible if l=b=h The cuboid is a cube its volume = l3=729 =>l=9. Its laternal surface area =4l2=324
Q. No. 10:
A solid right circular cone having slant heightL and radius r is cut from the vertex along the height h(from top to bottom) in two identical pieces. What is the ratio of the total surface area of any such piece to the total surface area of the original cone?
Answer: C The total surface area of cone is pie(rL+r2) After cutting it becomes pie(rL+r2)/2 each, which was exposed surface earlier also, plus a new vertical surface. But the new area that has cone as the cross section is a triangle, which has an area 1/2 *(2r)*h =rh So, the total surface area of each of the two pieces = pie(rL+r2)/2 +rh The ratio is 1/2 + h/[pie(r+L)]
Q. No. 11:
A cylinder is inscribed inside a cone having base radius 21 cm and height 21 cm. Find the maximum possible volume of the cylinder.
Answer: A Let the height of the cylinder =h => height of the cone just above the top circular surface of the cylinder =21-h Since that small cone and the original complete cone are similar. Height of the cone = Radius of small cone =21-h But, radius of small cone = radius of cylinder Thus volume of cylinder = pie * (21-h)2*h => V= pie/2 (21-h)2*(2h) Sum of all the variables terms in the above expression = (21-h)+(21-h)+2h=42 Thus product of these three terms would be maximum when all these would be maximum when all the variables terms are equal. V= pie/2 * (42/3)3= 4312 cm3
Q. No. 12:
A rectangular reservoir of 1000 cubic m capacity has length and height equal to 5m and 20 m respectively.If 200 cubic m of water is taken out of the reservoir, find the drop in the level of water (in meters) in the reservoir?
Answer: B Given that volume (V) = 1000 cubic m Length(l) = 5m and Height(h)= 20 m Breadth = V/lh = 10m The volume of water taken out = length x breadth x drop in the level of water => 200= 5*10*d => d=4 m