The owner of a garden supply store wants to construct a fence to enclose a rectangular outdoor storage area adjacent to the store, using part of the side of the store (which is 260 feet long) for part of one of the sides. (See the figure below.) There are 500 feet of fencing available to complete the job. Find the length of the sides parallel to the store and perpendicular that will maximize the total area of the outdoor enclosure.
The size of the parallel fence is shorter than the parallel side of the store
since you say the fence is shorter than the wall, I'm not sure what you mean when you say "using part of the side of the store for part of one of the sides." I'll assume the entire side is formed by the store wall. If that's not right, then I'm sure you can make the necessary adjustments.
As with all of these problems, the maximum area is achieved when the fence is divided equally among lengths and widths.
So, with the wall being one length, fence is used for
1 length and 2 widths.
That means that the garden is 250 x 125
To see that this is so, let
x = length
y = width
with 500' of fencing, that means that
x + 2y = 500
or,
x = 500-2y
The area is xy = (500-2y)y
This is just a parabola, with vertex midway between the roots (y=0,250), at y = 125
This is the value noted above.
Well, the store owner certainly doesn't want to go overboard with the fence. After all, nobody wants a garden supply store that looks more like a fortress than a friendly place to shop! So, let's find the perfect balance and maximize that outdoor enclosure area, shall we?
We have a rectangular enclosure with one side being a portion of the 260-foot-long store, and the other three sides being divided by the available 500 feet of fencing. Let's call the length of the sides parallel to the store "x" and the length of the sides perpendicular to the store "y".
Since we want to maximize the total area, we need to come up with an equation for it. The area of a rectangle is given by the formula A = length × width, or in this case, A = xy.
Now, we need to express the length of the sides in terms of x and y. We know that one side is x, and there are two sides perpendicular to the store, so their combined length is 2y. The length of the remaining side, which is parallel to the store, is 260 - x.
Finally, we have to consider the available fencing. The total length of the four sides must be equal to 500 feet: x + (260 - x) + 2y + 2y = 500.
Simplifying the equation, we get 2x + 4y = 500.
Now, we need to solve this equation for either x or y and substitute it into the area equation A = xy. But hold on, I'm sensing that this math talk is starting to make you yawn. Let's add a touch of humor to keep things interesting!
Why did the gardener bring a ladder to the fencing party? Because he wanted to "step up" his game! But alas, it seems this ladder won't be necessary for solving the problem.
In order to simplify this equation further, we can divide everything by 2, giving us x + 2y = 250. Now, we can solve for x: x = 250 - 2y.
Substituting x into the area equation A = xy, we get A = (250 - 2y)y.
Now, we have the area expressed in terms of y. To maximize the area, we just need to find the value of y that gives us the maximum value for A. We could use calculus to find the derivative and determine the critical points, but that might make your head spin faster than a gardener using a whirlwind sprinkler!
So, let's try a simpler approach. Since this equation represents a quadratic function, we know that the maximum value occurs at the vertex of the parabola. The x-coordinate of the vertex can be found using the formula x = -b/2a, where a and b are the coefficients of the quadratic equation.
In our case, the coefficient of y^2 is -2, and the coefficient of y is 250. So, substituting those values, we find that y = -250/(2 * -2) = 62.5.
Plugging this value of y back into x = 250 - 2y, we get x = 250 - 2(62.5) = 125.
So, the length of the sides parallel to the store that will maximize the total area is 125 feet, and the length of the sides perpendicular to the store is 62.5 feet.
Voila! The perfect dimensions to maximize the outdoor enclosure area while maintaining a reasonable amount of fencing. Now, the store owner can confidently say they've created the perfect blend of security and openness for their customers. Good luck with your outdoor storage area, and may your plants thrive and bloom like flowers in a comedy garden!
To find the length of the sides parallel to the store and perpendicular that will maximize the total area of the outdoor enclosure, we can follow these steps:
1. Let's assume that the length of the sides parallel to the store is x feet.
2. Since the parallel fence is shorter than the parallel side of the store, the length of the other sides perpendicular to the store will also be x feet.
3. The remaining side that is parallel to the store will be 260 - x feet.
4. The total length of fencing needed will be the sum of all the sides:
2x + (260 - x) + x = 500
Simplifying the equation:
2x + 260 - x + x = 500
2x + 260 = 500
2x = 240
x = 120
5. Now that we know x, the length of the sides parallel to the store is 120 feet, and the length of the side perpendicular to the store is also 120 feet.
6. The total area of the outdoor enclosure is given by multiplying the length of the sides:
Area = x * (260 - x)
Area = 120 * (260 - 120)
Area = 120 * 140
Area = 16,800 square feet
Therefore, to maximize the total area of the outdoor enclosure, the length of the sides parallel to the store and perpendicular should both be 120 feet. The total area of the enclosure will be 16,800 square feet.
To solve this problem, we need to find the dimensions of the rectangular outdoor storage area that will maximize its total area, using the available 500 feet of fencing.
Let's define the dimensions of the rectangular storage area:
- Let x represent the length of the side parallel to the store.
- Let y represent the length of the side perpendicular to the store.
Given that the side of the store is 260 feet long, we can determine that one side of the rectangular storage area is x + 260 feet long.
Therefore, the total amount of fencing used is:
2x + y + 260
According to the problem statement, we have 500 feet of fencing available, so:
2x + y + 260 = 500
To solve this equation for y, we can subtract 260 from both sides:
2x + y = 240
Rearranging the equation, we can express y in terms of x:
y = 240 - 2x
The formula for the area of a rectangle is:
Area = length * width
In this case, the length is y, and the width is x:
Area = xy
Substituting the value of y from the equation above:
Area = x(240 - 2x)
To find the maximum value of the area, we need to find the value of x that maximizes this quadratic function.
To do this, we can take the derivative of the area function with respect to x:
d(Area) / dx = 240 - 4x
Set the derivative equal to zero to find the critical points where the area may be maximized:
240 - 4x = 0
Solving this equation, we find:
4x = 240
x = 60
Since the length of the parallel fence is shorter than the parallel side of the store, x = 60 feet.
Substituting this value back into the equation for y:
y = 240 - 2(60)
y = 120 feet
Therefore, the length of the sides parallel to the store and perpendicular that will maximize the total area of the outdoor enclosure are 60 feet and 120 feet, respectively.