8 A rectangulaer garden with an area of 30m is Surrounded by a grass border Im wide on two Sides and am on tle other Side. what dimensions of the gerdon minimize the combined area of the garden and borders ?
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
![**Problem 8: Minimizing the Combined Area of a Garden and Borders**
A rectangular garden with an area of \(30 \, \text{m}^2\) is surrounded by a grass border. The grass border is 1 meter wide on two sides and 2 meters wide on the other two sides.
**Question:** What dimensions of the garden minimize the combined area of the garden and the borders?
This problem involves optimization, a common topic in calculus and geometry. The goal is to find the dimensions of the rectangular garden (length and width) that will minimize the total area, which includes both the garden and the surrounding grass borders.
### Explanation:
- Let the dimensions of the garden be \( l \times w \) where \( l \) is the length and \( w \) is the width.
- The area of the garden is given: \( l \times w = 30 \, \text{m}^2 \).
Surrounding this garden is a grass border:
- On two sides, the width of the border is 1 meter.
- On the other two sides, the width is 2 meters.
To define the total area (garden + borders):
- The total length considering the borders would be \( l + 2 \times 2 = l + 4 \) meters.
- The total width considering the borders would be \( w + 1 + 1 = w + 2 \) meters.
Thus, the combined area \(A_{total}\) is:
\[ A_{total} = (l + 4)(w + 2) \]
Given the area of the garden (\( l \times w = 30 \)), we can set up the equation to minimize:
\[ A_{total} = (l + 4)(\frac{30}{l} + 2) \]
The task then involves differentiating this equation with respect to \( l \) and finding the value of \( l \) that minimizes \( A_{total} \). We then use this \( l \) to find the corresponding \( w \) that satisfies \( l \times w = 30 \).
This kind of problem is frequently solved using techniques from calculus, such as finding the derivative, setting it to zero, and solving for the critical points.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1e9d947d-304f-415e-95bf-140f5d3b17e4%2Faa455b50-c852-44f6-b9fd-4a4210bb3a13%2Frr40vxn_processed.jpeg&w=3840&q=75)
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