A farmer wants to enclose three sides of a rectangular pasture unsing 1000 yards of fencing. The fourth side does not require fencing because it borders a river.
What dimensions (length and width) should the farmer choose in order to enclose the greatest area?
a) Find at least five ordered pairs, comparing pasture length to its area. Remember: only 1000 yards of fencing available.
(Don't i need the maximum to determine this?)
b) Plot your five data points (length and area) on a grid with scales, and join them roughly. You should get the shape of a parabola.
c) Use your graping calculator to find the quadratic function that related to the data.
d) Graph resulting function by hand. State coordinateds of the point of the graps that yields max area. Find max area the farmer can fence in. What are the dimensions?
Can someone help me with this. I am so lost.
All Help is appreciated.
Considering all rectangles with a given perimeter, one side being provided by a straight given boundry, which one encloses the largest area?
Letting P equal the given perimeter and "x" the short side of the rectangle, we can write for the area A = x(P - 2x) = Px - 2x^2.
Taking the first derivitive and setting equal to zero, dA/dx = P - 4x = 0, x becomes P/4.
With x = P/4, we end up with a rectangle with side ratio of 2:1.
.....The short side is P/4.The traditional calculus approach would be as follows.
.....The long side is (P - 2(P/4)) = P/2.
Therefore, it can be unequivicably stated that of all possible rectangles with a given perimeter, one side being a given external boundry, the rectangle with side ratio of 2:1 encloses the maximum area.
The prime factorization os 1000 is 2^3(5^3).
Therefore, the total number of factors(divisors) is F = (3 + 1)(3 + 1) = 16, namely 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 10, 125, 250, 500 and 1000 enabling 8 possible sets of dimensions.
500
Sure! I can walk you through the steps to solve this problem.
a) To find the dimensions that will enclose the greatest area, we need to experiment with different lengths and widths of the rectangular pasture. Let's start by finding at least five ordered pairs of length and area.
Length (yards) | Width (yards) | Area (square yards)
-------------------------------------------------------
100 400 40000
200 300 60000
300 200 60000
400 100 40000
500 0 0
By using these dimensions, we ensure that the three sides require 1000 yards of fencing.
b) Now, let's plot these five data points on a graph. On the x-axis, we will have the length (yards), and on the y-axis, we will have the area (square yards). Joining these points will give us the shape of a parabola.
c) To find the quadratic function that relates to this data, we can use a graphing calculator. By inputting the ordered pairs into the calculator, it will generate the best-fit quadratic function for the data.
d) After obtaining the quadratic function, we can graph it by hand. By exploring the shape of the graph, we can identify the coordinates of the point that yields the maximum area. The maximum area will correspond to the vertex of the parabola.
By finding the vertex, we can determine the dimensions (length and width) that will enclose the greatest area. The maximum area will be the y-coordinate of the vertex.
Let's start with step a) and find the ordered pairs.
Sure, I can guide you through the steps to solve this problem.
a) To find at least five ordered pairs comparing pasture length to its area, you need to start by understanding the constraints given in the problem. The problem states that there are 1000 yards of fencing available and that the fourth side of the pasture does not require fencing. So, you can express the perimeter of the rectangular pasture using the length and width.
Let's denote the length of the pasture as L and the width as W. Since the perimeter of the rectangular pasture is made up of three sides, you can set up the equation:
2L + W = 1000
Now, you need to express the area of the pasture in terms of L and W. The area of a rectangle is given by the formula:
Area = L * W
b) To plot the five data points, you can start by picking random values for L and then use the equation 2L + W = 1000 to solve for W. This will give you pairs of (L, W), and you can calculate the corresponding areas using the formula Area = L * W.
For example, you can start with L = 100 and solve for W:
2(100) + W = 1000
W = 800
So, one of the data points would be (100, 800), since the length is 100 and the width is 800, resulting in an area of 80,000 square yards.
Repeat this process four more times to get a total of five data points.
c) Once you have your five data points, you can plot them on a graph with scales and roughly join them. This will give you the shape of a parabola.
d) To find the quadratic function that relates to the data, you can use a graphing calculator. Enter the data points and let the calculator determine the quadratic equation that best fits the data.
Once you have the quadratic equation, you can graph it by hand. Locate the coordinates of the point on the graph that yields the maximum area. This point will give you the dimensions (length and width) for which the farmer can enclose the greatest area.
To find the maximum area, substitute the values of the coordinates into the area formula: Area = L * W.
I hope this explanation helps you understand how to approach the problem. Let me know if you have any further questions!