Question

a gardener has 120 feet of fencing. he wishes to enclose all four sides of a rectangular area with this fencing. a quadratic function, giving the area (a) of these rectangles, that models this problem situation is a(x) = 60x − x², where x represents the length of the rectangle. what is a reasonable domain and range for this area function?
a. domain = (0, 120) range = (0, 60)

b. domain = (0, 30) range = (0, 225)

c. domain = (0, 60) range = (0, 900)

d. domain = (0, 120) range = (0, 3600)

Explanation:

Step1: Analyze the domain

The gardener has 120 feet of fencing for a rectangle, so the perimeter \( P = 2(x + y)=120 \), where \( x \) is length and \( y \) is width. Simplifying, \( x + y = 60 \), so \( y = 60 - x \). Since length and width must be positive, \( x>0 \) and \( y = 60 - x>0\Rightarrow x < 60 \). Thus, the domain of \( x \) is \( (0, 60) \).

Step2: Analyze the range

The area function is \( A(x)=60x - x^{2} \), which is a quadratic function opening downward (coefficient of \( x^{2} \) is -1). The vertex of \( ax^{2}+bx + c \) is at \( x = -\frac{b}{2a} \). Here, \( a=-1 \), \( b = 60 \), so vertex \( x=-\frac{60}{2(-1)} = 30 \). Substituting \( x = 30 \) into \( A(x) \), we get \( A(30)=60(30)-30^{2}=1800 - 900 = 900 \). The minimum area approaches 0 (as \( x \) approaches 0 or 60). So the range is \( (0, 900) \).

Answer:

C. domain = (0, 60), range = (0, 900)