A rectangular field is to have an area of 500 sq m.
a) write the perimeter, P, of the field as a function of the width, w.
b) find the domain of the function in part a.
c) use a graphing calculator to sketch the graph of the function in part a.
d) describe what the graph found in a part c tells you about how the perimeter of the field varies with the width
a) To write the perimeter, P, of the field as a function of the width, w, we need to consider the formula for the perimeter of a rectangle. The formula is P = 2L + 2W, where L represents the length and W represents the width of the rectangle. In this case, we are given that the area of the field is 500 sq m.
To determine the length, we can use the formula for the area of a rectangle, which is A = L * W. Since we are given that the area is 500 sq m, we can write the equation as 500 = L * W.
Solving this equation for L, we get L = 500 / W.
Substituting this value for L in the equation for the perimeter, we have P = 2(500 / W) + 2W.
Therefore, the perimeter, P, of the field can be written as a function of the width, w, as P(w) = 1000 / w + 2w.
b) To find the domain of the function P(w), we need to consider any restrictions on the width, w, of the field. In this case, there are no stated restrictions or limitations on the width. Therefore, the domain of the function P(w) is all real numbers except for w = 0, since division by zero is undefined.
c) To sketch the graph of the function P(w) = 1000 / w + 2w on a graphing calculator, follow these steps:
1. Turn on the graphing calculator and select the graphing function.
2. Enter the equation P = 1000 / w + 2w.
3. Specify the viewing window by setting appropriate values for the x-axis (width) and y-axis (perimeter) range.
4. Plot the graph.
d) The graph found in part c tells us how the perimeter of the field varies with the width. As we can see from the graph, the relationship between the width and the perimeter is not a simple linear relationship. The graph appears to be a hyperbola opening upward. As the width, w, increases, the perimeter, P, decreases. Conversely, as the width, w, decreases, the perimeter, P, increases. This is because as the width of the rectangle increases, the length decreases in order to maintain the fixed area of 500 sq m. Therefore, the graph visualizes the inverse relationship between the width and the perimeter of the rectangular field.