Calculus- area of a garden problem

In summary, to find the dimensions of the rectangular rose garden that will minimize the cost of the surrounding brick wall and fence, you need to write a formula for the total cost, which will be a function of the length and width of the garden. By minimizing this cost function, you will be able to determine the optimal dimensions for the garden.
  • #1
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Homework Statement


A rectangular rose garden will be surrounded by a brick wall on three sides and a fence on the fourth side. The area of the garden will be 1000m2. The cost of the brick wall is $192/m. The cost of the fencing is $48/m. Find the dimensions of the garden so that the cost is a minimum. [domain and test not needed]


Homework Equations



So I know A = 1000m2 and A=lw

The Attempt at a Solution


My question is, how do I do this when the sides are different due to the materials used in building them? It's not like a question where you fence in three sides with the same stuff and just optimize costs...I'm very slow at calculus, please explain details! I really appreciate this.
 
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  • #2
You want to think about the perimeter and its cost. Start by writing a formula for the cost:

Cost = some function of L and W.

That's what you want to minimize.
 

1. What is the purpose of using calculus to solve a garden area problem?

Calculus is a branch of mathematics that deals with the study of change and motion. It is used to solve problems involving rates of change, such as finding the area of a garden that is constantly changing in size and shape.

2. What are the key concepts of calculus that are applied in solving a garden area problem?

The two key concepts of calculus used to solve a garden area problem are differentiation and integration. Differentiation is used to find the rate of change of the area of the garden, while integration is used to find the total area of the garden by adding up all the small changes in area over time.

3. How does calculus help in finding the optimal size and shape of a garden?

Calculus allows us to find the maximum or minimum value of a function, which is useful in determining the optimal size and shape of a garden. By finding the maximum area of a garden, we can determine the ideal dimensions for the garden to achieve the largest possible area.

4. Can calculus be used to solve complex garden area problems?

Yes, calculus can be used to solve complex garden area problems. It allows us to find the area of irregularly shaped gardens, as well as gardens with changing dimensions over time. Additionally, calculus can be used to optimize the area of a garden by finding the maximum or minimum value of a function.

5. Are there any real-world applications of using calculus to solve garden area problems?

Yes, there are many real-world applications of using calculus to solve garden area problems. Calculus is commonly used in landscaping and gardening to determine optimal garden sizes and shapes. It is also used in agriculture to maximize crop yields and in architecture to design gardens with specific dimensions and shapes.

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