poul has 124.8m of encing to build a rectangular pen. He wants the pen to have the largest possible area. Calculate the length and width of the pen to one decimal place.
I don't get it at all
P = 2 L + 2W
124.8 / 4 = 31.2
The largest rectangular pen he can build is a square.
(a square is a rectangle)
so divide 124.8 by 4 to get the length of each side.
thanks
To solve this problem, we need to use the concept of optimization. In this case, we want to maximize the area of a rectangular pen given a fixed amount of fencing.
Let's break down the problem step by step:
1. Recall the formula for the perimeter (P) of a rectangle: P = 2(length + width)
2. Since Poul has 124.8m of fencing, we can write the equation as follows: 124.8 = 2(length + width)
3. We can rearrange the equation to solve for either length or width. Let's solve for length: length = (124.8 - 2width)/2
4. Now, we need to express the area (A) of the rectangular pen in terms of the width. The area of a rectangle is given by: A = length * width
5. Substitute the expression for length from step 3 into the area formula: A = ((124.8 - 2width)/2) * width
6. Simplify the equation: A = (62.4 - width) * width
7. To find the maximum area, we need to find the width that maximizes the area. We can do this by taking the derivative of the area function with respect to width and setting it equal to zero.
8. Differentiate the area function using the power rule: dA/dwidth = 62.4 - 2width
9. Set dA/dwidth equal to zero and solve for width: 62.4 - 2width = 0
10. Simplify and solve: 2width = 62.4, width = 31.2
11. Now that we have the width, we can substitute it back into the equation for length to find its value: length = (124.8 - 2(31.2))/2 = (124.8 - 62.4)/2 = 62.4/2 = 31.2
Therefore, the length and width of the rectangular pen that maximize its area while using 124.8m of fencing are both 31.2 meters.