# Gr 12 Calc: Optimization problem/min cost

• Specter
In summary, the homeowner wants to enclose a rectangular garden with fencing on all four sides, with the shared fence being paid for by the neighbor. The area of the garden is 432 m^2 and the homeowner wants to minimize their cost. A diagram is provided to help visualize the problem. To find y, the height of the garden, the homeowner must find the side of the fence that is shared with the neighbor. Then, a cost function can be set up using the price per meter to calculate the total cost for both the homeowner and the neighbor.
Specter

## Homework Statement

A homeowner wishes to enclose a rectangular garden with fencing. The garden will be adjacent to his neighbour’s lot. There will be fencing on all four sides. His neighbour will be paying for half the shared fence
1. What should the dimensions of the garden be if the area is 432 m^2 and the homeowner would like to keep his share of the cost to a minimum? Include a diagram.
2. At \$45/m, what is the homeowner’s cost? What is the neighbour’s cost?

## The Attempt at a Solution

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My diagram https://i.imgur.com/4Pvpxg3.jpg

From there I know that I need to find what y equals but this is where I get stuck. After I find y I need to find
the side of the fence that is shared with the neighbour. Then I can set up a cost function. I haven't done any questions like this before so I am struggling. I guess I just need some help setting up the problem. Thanks

Specter said:
Then I can set up a cost function
Make an attempt... you know the price per meter, you want to know the cost, so you'll need ...

## 1. What is an optimization problem?

An optimization problem is a mathematical problem where the goal is to find the best possible solution from a set of feasible options. In other words, it involves finding the maximum or minimum value of a function given a set of constraints.

## 2. What is the purpose of optimization problems in calculus?

Optimization problems are commonly used in calculus to find the maximum or minimum value of a function, which can be applied to real-world situations. For example, a business may use optimization to determine the most cost-effective production level for a product.

## 3. How do you solve an optimization problem?

To solve an optimization problem, you first need to identify the objective function and the constraints. Then, you can use techniques such as taking derivatives and setting them equal to zero, graphing, or using algebraic methods to find the optimal solution.

## 4. What is the difference between maximum and minimum optimization problems?

In a maximum optimization problem, the goal is to find the highest possible value of the objective function. In a minimum optimization problem, the goal is to find the lowest possible value. The approach to solving these problems may differ, but the basic principles and techniques used are the same.

## 5. How do you apply optimization to real-world situations?

Optimization can be applied to real-world situations by formulating the problem into a mathematical model and solving it using calculus techniques. This can help individuals and businesses make informed decisions and find the most efficient and cost-effective solutions to various problems.

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