Topic : Area of Circle.

Question : A circle with area A1 is contained in the interior of larger circle with area A1+A2. If the radius of the larger circle is 3 and A1,A2,A1+A2 are in A.P,then the radius of smaller circle is__________.

Answer :

radius of the larger circle = 3
area of the larger circle = A1 + A2

A1+A2 = pi r^2 = 9 pi

A1+A2 = 9 pi —-(1)

A1, A2, A1+A2 are in A.P

hence the difference is common

A2 -A1 = A1 + A2 – A2 = A1

A2 = 2A1 —(2)
solving (1) and ( 2) ,

3A1 = 9 pi -> A1 = 3pi

Area of the smaller circle = 3 pi

radius of the smaller circle = square root of 3

(we have two concentric circles ; A1is the area of the inner circle , A2 is the area between the inner and outer circle)

Topic : parabola

Question : Growth rate y (in pounds/month)of an infant is related to the present weight x(in pounds)by formula y= cx(21-x)where c is a positive constant.0<x<21.At what rate does the maximum growth occur.

Answer :

The y coordinate of the vertex is the max of a quadratic if the parabola is downward and its the minimum if its upward.

the parabola is upward if the co efficient of x^2 is positive and downward if its negative.

here we have y= c x(21-x)= 21cx -c x^2

as the co efficient of x^2 is negative, we get a maximum value

formula: for y = ax^2+bx+c , the vertex is given by ( -b/2a , f( -b/2a) )

hence the maximum y value is -b/2a

here the max growth rate = – 21c / -2c = 21/2 =10.5pounds/month.