APPLICATION OF CONTINUITY OF A FUNCTION

Question

1.   A rancher has 280 feet of fencing material to enclose two adjacent pastures as shown in the figure.

a.   Write the total area  of the two pastures as a function of x.

b.   Identify the domain of the function.

c.   Show that the function is continuous at its domain.

Answer 1

2y = 280 -3x

y = 140 -3x/2

total area = 2xy = 2x(140-3x/2) = 280x - 3x^2

A = 280x -3x^2

domain is 0<x< 93 1/3

maximum total area is when

A' = 280-6x = 0

x = 280/6 = 140/3 = 46 2/3 ft and y= 140-(3/2)(140/3) = 70 feet

max A = 140(140/3) = 6533 1/3 ft^2

range is 0<A(x)<6533 1/3

A is a downward opening parabola with x intercepts (0,0) and (93 1/3, 0), vertex (46 2/3, 6,533 1/3)

the area can be any real number between zero and 6,533 1/3, as x varies from zero to 93 1/3 feet