A farmer wants to construct a pen with 100 feet fencing. The pen will be divided into two areas. Find the dimensions of the big pen. You may use your calculator or the fact that the x-coordinate of the vertex of a parabola is x=-b/2a.
I would like to show a picture but somehow it won't allow to download. The image is a rectangle divided into 2 equal sections by a line. width is x and their is 3x and y is a length
3x + 2y = 100
a = xy = x(100-3x)/2 = (100x - 3x^2)/2
max area at x = 100/6
Note that the fencing is divided equally among widths and lengths for max area.
To find the dimensions of the big pen, let's break down the problem step by step:
1. Start by visualizing the problem. We have a rectangular pen divided into two equal sections by a line. One section has a width of 'x,' and the other section also has a width of 'x.' The length of both sections is 'y.'
2. We are given that the total amount of fencing available is 100 feet. Since the fencing goes around the entire pen, we need to consider the sum of all the sides.
3. For the big pen, we have two lengths and three widths. The lengths will be the same as the individual sections, so both will be 'y.' The widths will be the sum of both individual sections' widths, so we have 2x.
4. To set up an equation based on the given information, we can use the perimeter formula of a rectangle: P = 2L + 2W, where P is the perimeter, L is the length, and W is the width.
Applying this formula to our problem, we have:
100 = 2y + 2(2x)
5. Simplify the equation:
100 = 2y + 4x
6. Rearrange the equation to get it into a more useful form:
2y = 100 - 4x
y = 50 - 2x
7. We can use the fact that the x-coordinate of the vertex of a parabola is x = -b/2a to find the value of 'x' that maximizes 'y.'
8. By comparing the equation y = 50 - 2x to the general form y = ax² + bx + c, we can see that 'a' is -2, 'b' is 0, and 'c' is 50.
9. Applying the formula x = -b/2a, we get:
x = -0 / (2 * (-2))
x = 0
10. Since 'x' cannot be negative (as it represents a dimension in the pen), we know that the dimensions of the big pen will be:
Width: 2x = 2 * 0 = 0 feet
Length: y = 50 - 2(0) = 50 feet
Therefore, the dimensions of the big pen are a width of 0 feet and a length of 50 feet.