Westonci.ca offers quick and accurate answers to your questions. Join our community and get the insights you need today. Discover comprehensive solutions to your questions from a wide network of experts on our user-friendly platform. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
Answer:
A rectangular fence of width [tex]600[/tex] feet and length [tex]600[/tex] feet would achieve the maximum possible area: [tex]360\, 000[/tex] square feet.
Step-by-step explanation:
Let [tex]x[/tex] denote the width of the fence. These would account for two of the sides of the rectangle. There would be [tex](2400 - 2\, x)[/tex] of the fence remaining. Since these need to be evenly split between the other two sides of the rectangle, the "length" of the fence would need be [tex](1/2)\, (2400 - 2\, x) = 1200 - x[/tex].
The area of the rectangle would be the product of length and width:
[tex](x) (1200 - x) = -x^{2} + 1200\, x[/tex].
Note that since both width [tex]x[/tex] and length [tex](1200 - x)[/tex] need to be non-negative, [tex]0 < x < 1200[/tex].
When finding value of [tex]x[/tex] that would maximize or minimize the value of a continuous function, such as the area [tex](-x^{2} + 1200\, x)[/tex], within a given interval, the value of [tex]x[/tex] might be:
- Within the range of the interval, which requires the first derivative test, or
- If the interval is not an open interval: at the endpoints of the interval, which requires evaluating the function at these points.
Since in the given interval [tex]0 < x < 1200[/tex] , [tex]x[/tex] cannot be equal to either endpoint [tex]0[/tex] or [tex]1200[/tex], this interval would be an "open interval". It would not be necessary to evaluate the function at these two endpoints.
To find the values of [tex]x[/tex] that would maximize the area [tex](-x^{2} + 1200\, x)[/tex] within the range of the interval apply the first derivative test:
- Differentiate the expression for area with respect to [tex]x[/tex] to obtain the first derivative. Find values of [tex]x[/tex] that would set the first derivative to [tex]0[/tex].
- Evaluate the expression for area at each of these values of [tex]x[/tex]. Find the value of [tex]x[/tex] that achieves the maximum possible value.
The first derivative of area [tex](-x^{2} + 1200\, x)[/tex] with respect to width [tex]x[/tex] is:
[tex]\displaystyle \frac{d}{d x}\left[-x^{2} + 1200\, x\right] = -2\, x + 1200[/tex].
Solve for values of [tex]x[/tex] that would set this first derivative to [tex]0[/tex]::
[tex]-2\, x + 1200 = 0[/tex].
[tex]x = 600[/tex].
Evaluate the expression for area at this point:
[tex]-x^{2} + 1200\, x = -(600)^{2} + (1200)\, (600) = 360\, 000[/tex].
The length of the rectangular fence would be [tex](1200 - x) = 600[/tex].
In other words, a rectangular fence [tex]600[/tex] feet in width and [tex]600[/tex] feet in length would achieve the maximum possible area: [tex]360\, 000[/tex] square feet.
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.
