a farmer has 400 meters of fence with which to enclose a portion of land. the farmer wants to enclose a rect angular piece of ground that is as large as possible. the land is bordered by water on two sides.
There are three options for the farmer
Option 1) have two sides bordered by the water and two sides by the fence
Option 2) have one side bordered by water and three sides by the fence
Option 3) use fence for all four sides
Construct formula for the area in each of the three options( hint: name one side x)
c) explain which option is best, and why?
Someone help, i got the formulas but idk if they are correct
100
To find the formulas for the area in each of the three options, we need to break down the problem:
Option 1:
In this case, two sides are bordered by the water, which means the length of the rectangle will be determined by the 400 meters of fence available. Let's call the length of the rectangle as 'x' and the width as 'y'. The perimeter is given by 2x + y = 400. Since two sides are bordered by water, the equation becomes: x + y = 400. The area is given by A = x * y.
Option 2:
In this case, one side is bordered by water, so the length of the rectangle, which we'll call 'x', will be determined by the available fence. Let's call the width of the rectangle as 'y'. The perimeter is given by x + 2y = 400. Since one side is bordered by water, the equation becomes: x + 2y = 400. The area is given by A = x * y.
Option 3:
In this case, all four sides of the rectangle are enclosed by the fence, so both the length and width will be determined by the fence. Let's call the length of the rectangle as 'x' and the width as 'y'. The perimeter is given by 2x + 2y = 400. Since all four sides are enclosed by the fence, the equation remains as 2x + 2y = 400. The area is given by A = x * y.
Now, let's determine which option is best.
To find the largest area, we need to maximize the value of A = x * y.
Option 1:
The perimeter equation x + y = 400 can be rearranged to y = 400 - x. By substituting this value in the area equation, we have A = x * (400 - x) = 400x - x^2. To find the maximum, we can take the derivative of A with respect to x, set it equal to zero, and solve for x. Once we have the value of x, we can find y using y = 400 - x. This will give us the dimensions of the rectangle that maximize the area.
Option 2:
The perimeter equation x + 2y = 400 can be rearranged to y = (400 - x)/2. By substituting this value in the area equation, we have A = x * (400 - x)/2 = (400x - x^2)/2. Similar to option 1, we can take the derivative of A with respect to x, set it equal to zero, and solve for x. Once we have the value of x, we can find y using y = (400 - x)/2.
Option 3:
In this case, since both the length and width of the rectangle are determined by the fence, the area A = x * y. We can maximize the area by finding the dimensions of the rectangle that maximize the product of x and y.
To determine which option is best, we need to compare the maximum areas obtained from each option. From the three formulas derived above, calculate the maximum areas by solving for the values of x and y using the methods mentioned.
Compare the maximum areas obtained from each option, and the one with the largest area will be the best option for the farmer to use.