GMAT Question 15-17

Q-15.
A conical tent is to accommodate 10 persons. Each person must have 6 meter square space to sit and 30 meter cube of air to breathe. What will be the height of the cone?

Solution:
Given that each person should have a 6 m.sq. of area, that means the base should have a total area of 6*10 = 60 m.sq. (which is pi*r^2, r being the radius of the base)

  The volume of the cone will have to be 30*10 m.cube. as each person needs 30 m.cube volume. Thus using the expression for the volume of a cone  1/3 pi*r^2*h = 300

Now pi*r^2 = 60, thus the above expression becomes (1/3)*60*h=300 => h=15 m.

Q-16.
If the curved surface area of a cone is thrice that of another cone and slant height of the second cone is thrice that of the first cone, find the ratio of the area of their base.

Solution:
Curved SA of a cone=pi*r*L  let r1 and l1 for 1st cone  and r2 and l2 for 2nd  Csa of 1st=3( Csa of2nd)  pi*r1*l1=3pi*r2*l2=>r1l1=3r2l2   l2=3l1  Therefore r1*l1=9r2*l2  i.e r1=9r2   base=pi r^2  ratio is pi r1^2/pi r2^2=> 81:1

Q-17.
A thin piece of wire 40 meters long is cut into two pieces. One piece is used to form a circle with radius r, and the other is used to form a square. No wire is left over. Which of the following represents the total area, in square meters, of the circular and square regions in terms of r?  1) πr^2  2) πr^2 + 10  3) πr^2 + 1/4π^2r^2  4) πr^2 + (40 - 2πr)^2  5) πr^2 + (10 - 1/2πr)^2

Solution:
Area of the circle = πr^2 
Area of the square = one of its sides squared 
Perimeter of the square is 40 - (the perimeter of the circle = 2πr) 
One of the sides of the square = 40-2πr/4 or 10-1/2πr 
Total area: πr^2 + (10-1/2πr)^2 (E)