Jenna's Rectangular Garden Borders A Wall. She Buys 80 Ft Of Fencing. What Are The Dimensions Of The Garden That Will Maximize Its Area?A. ${ 5'\times 70'\$} B. ${ 3'\times 14'\$} C. ${ 15'\times 50'\$} D. [$20'\times
Maximizing the Area of a Rectangular Garden
In this problem, we are tasked with finding the dimensions of a rectangular garden that will maximize its area, given a fixed amount of fencing. This is a classic optimization problem in mathematics, and it can be solved using calculus. We will use the concept of optimization to find the dimensions of the garden that will result in the maximum area.
Jenna has 80 ft of fencing to border a rectangular garden. She wants to know the dimensions of the garden that will maximize its area. To solve this problem, we need to find the dimensions of the garden that will result in the maximum area, given the constraint of 80 ft of fencing.
Let's denote the length of the garden as L and the width as W. The perimeter of the garden is given by the equation:
P = 2L + 2W
Since Jenna has 80 ft of fencing, we can set up the equation:
2L + 2W = 80
We can simplify this equation by dividing both sides by 2:
L + W = 40
To maximize the area of the garden, we need to find the values of L and W that will result in the maximum area. The area of the garden is given by the equation:
A = LW
We can use calculus to find the maximum area. We will take the derivative of the area equation with respect to L and set it equal to zero:
dA/dL = W + L(dW/dL) = 0
Since W is a function of L, we can substitute W = 40 - L into the equation:
dA/dL = (40 - L) + L(d(40 - L)/dL) = 0
Simplifying the equation, we get:
dA/dL = 40 - 2L = 0
Solving for L, we get:
L = 20
Now that we have found the value of L, we can find the value of W by substituting L = 20 into the equation W = 40 - L:
W = 40 - 20 W = 20
Therefore, the dimensions of the garden that will maximize its area are 20 ft by 20 ft. This is the only solution that satisfies the constraint of 80 ft of fencing.
The correct answer is:
A. ${20'\times 20'\$}
The other options are not correct because they do not satisfy the constraint of 80 ft of fencing. For example, option B has a perimeter of 2(3) + 2(14) = 60, which is less than 80 ft. Option C has a perimeter of 2(15) + 2(50) = 130, which is greater than 80 ft. Option D has a perimeter of 2(20) + 2(30) = 100, which is also greater than 80 ft.
The final answer is A. ${20'\times 20'\$}.
Maximizing the Area of a Rectangular Garden: Q&A
In our previous article, we discussed how to maximize the area of a rectangular garden given a fixed amount of fencing. We used calculus to find the dimensions of the garden that will result in the maximum area. In this article, we will answer some common questions related to this problem.
A: The main constraint in this problem is the fixed amount of fencing, which is 80 ft.
A: To find the dimensions of the garden that will maximize its area, we need to use calculus. We take the derivative of the area equation with respect to the length of the garden and set it equal to zero. This will give us the value of the length that will result in the maximum area.
A: The relationship between the length and width of the garden is given by the equation L + W = 40, where L is the length and W is the width.
A: To find the value of the width of the garden, we can substitute the value of the length into the equation W = 40 - L.
A: The dimensions of the garden that will maximize its area are 20 ft by 20 ft.
A: This is the only solution that satisfies the constraint of 80 ft of fencing because any other combination of length and width will result in a perimeter that is either less than or greater than 80 ft.
A: Yes, we can use this method to find the dimensions of a garden with a different amount of fencing. We simply need to substitute the new amount of fencing into the equation L + W = 40 and solve for the length and width.
A: Some real-world applications of this problem include designing gardens, parks, and other outdoor spaces. It can also be used to optimize the layout of buildings and other structures.
In this article, we answered some common questions related to maximizing the area of a rectangular garden given a fixed amount of fencing. We hope that this article has been helpful in understanding this problem and its applications.
The final answer is A. ${20'\times 20'\$}.