A gardener has 120 ft of fencing to fence in a rectangular garden. one side of the garden is bordered by a river and so it does not need any fencing.
a. what dimensions would guarantee a garden with an area of 1350 ft ^2
b. what dimensions would guarantee the greatest area? how much is the greatest area?
If the dimensions are x and y,
2x+y=120
xy = 1350
Solve that to get x and y. There are two possible solutions.
Note that the area is
xy = x(120-2x) = 120x-2x^2
That's just a parabola. The vertex represents the maximum area.
To solve this problem, we can use the formula for the area of a rectangle: A = length × width.
a. To guarantee a garden with an area of 1350 square feet, we need to find the corresponding dimensions of the rectangle.
Let's assume the length of the garden is x feet. Since one side is bordered by a river and doesn't need fencing, we only need to fence three sides. Two sides with lengths x feet and one side with length y feet.
We know that the perimeter of the garden is 120 feet, which means the sum of the lengths of the three sides is 120 feet:
x + x + y = 120
Simplifying the equation gives:
2x + y = 120
To find the area, we multiply the length and width:
A = x × y
Given that A = 1350 square feet, we can substitute the value into the equation:
1350 = x × y
Now we have a system of equations with two variables. We can solve it by substituting the value of y from the first equation into the second equation:
y = 120 - 2x
1350 = x × (120 - 2x)
By simplifying and rearranging the equation, we get a quadratic equation:
2x^2 - 120x + 1350 = 0
Using the quadratic formula, we can find the value of x:
x = (-b ± √(b^2 - 4ac))/(2a)
Here, a = 2, b = -120, and c = 1350. Substituting these values into the formula will give us two possible values for x.
Once we know the value of x, we can substitute it back into the equation y = 120 - 2x to find the corresponding value of y.
b. To find the dimensions that guarantee the greatest area, we need to find the maximum value of the area function A = x × y. Since we already know the equation for y in terms of x (y = 120 - 2x), we can substitute it into the area equation:
A = x × (120 - 2x)
Now, we can maximize the area by finding the critical points of the function A(x). We can do this by taking the derivative of A(x) with respect to x and setting it equal to zero:
dA/dx = 0
Then, we solve the resulting equation to find the critical value(s) of x. Substituting this value back into the equation y = 120 - 2x will give us the corresponding value of y.
Finally, we can substitute the values of x and y into the area equation A = x × y to calculate the greatest area.