To complete the statements regarding the scales for the axes of the function
-
For the horizontal axis (d):
- The variable
represents the width, and it must be a positive value. Since Peter has 250 feet of fencing, the maximum width can be is 125 feet (as must still be positive). A reasonable scale for the horizontal axis would allow us to visualize a range from 0 to 125 feet effectively. - Considering the options: 1 ft, 2 ft, 10 ft, or 75 ft, a scale of 10 ft would be reasonable, allowing for clear visualization of the width increments without making the graph too crowded.
- The variable
-
For the vertical axis (A(d)):
- The area
can be calculated as follows: - To find the maximum area, we can analyze the function. The area is a quadratic function, and its maximum value occurs at the vertex. The vertex
value can be calculated as: - Substituting back into the area function:
- The area goes to 0 when
or . Thus, the range for the area is from 0 to approximately 7812.5 square feet. - Considering the options for the vertical axis: 75 sq ft, 100 sq ft, 500 sq ft, or 2000 sq ft, a scale of 500 sq ft would be appropriate and manageable based on the maximum area.
- The area
Combining these observations, we can complete the statements:
A reasonable scale for the horizontal axis of the function