Question

mr. brown owns a ranch and is planning to fence off a rectangular pasture for his horses. he has enough materials to build 240 yards of fence. if one side of the pasture is x yards long, mr. brown figures out that the area of the pasture, in square yards, can be described by this function:
$y = -(x - 60)^2 + 3,600$
which property of the function tells you the greatest area mr. brown can fence off?

• the function’s zeros
• the function’s minimum value
• the function’s maximum value
• the function’s y-intercept

what is the greatest area mr. brown can fence off?
square yards

Explanation:

Step1: Analyze the function form

The function is given as \( y = -(x - 60)^2 + 3600 \), which is a quadratic function in vertex form \( y = a(x - h)^2 + k \), where \( a=-1 \), \( h = 60 \), and \( k = 3600 \).

Step2: Determine the property for maximum area

For a quadratic function \( y = a(x - h)^2 + k \), if \( a<0 \), the parabola opens downward, and the vertex \((h,k)\) represents the maximum point. Here, \( a=-1<0 \), so the function has a maximum value at \( k = 3600 \). The \( y \)-value of the function represents the area of the pasture, so the maximum value of the function gives the greatest area Mr. Brown can fence off.

Step3: Answer the first sub-question

The property that tells the greatest area is the function's maximum value.

Step4: Find the greatest area

From the vertex form, the maximum value of the function \( y = -(x - 60)^2 + 3600 \) is \( 3600 \) (since the vertex's \( y \)-coordinate \( k \) is the maximum value for a downward - opening parabola).

Answer:

• Which property: the function's maximum value
• Greatest area: \( 3600 \) square yards