Homework SourseA rectangle has one corner in quadrant I on the graph of y 3
A rectangle has one corner in quadrant I on the graph of y =36x^2, another at the origin, a third on the positive y-axis, and the fourth on the positive x-axis (a) Express the area A of the rectangle as a function of x. Simplify b) What is the domain of A? (Interval Notation) c) Graph A=A(x). For what value of x is A largest? (Round 2 Decimal places).
Solution
one corner in quadrant I on the graph of y =36x^2
Other point (0,0)
third on the positive y-axis,
and the fourth on the positive x-axis
lenght of side 1 = (0, y) - ( 0,0) = y
lenght of other side = (x, 0) -(0, 0) = x
Area of rectangle = lenght *breadth = y*x
substitute y = 36- x^2
a) Area (A) = ( 36 -x^2)(x) = 36x -x^3
b) A(x) = 36x -36x^3 is a polynomial function , it can take all real values of x
So, Domain = ( -inf, inf)
c) Largest area: find A\'(x) = 36 - 3x^2 =0
x^2 = 12 ----> x= +sqrt12
x = sqrt(12) = 3.46
