Question
What are the dimensions of a rectangular garden with an area of 100 sq ft that minimize the cost of materials for a brick wall costing $15 per foot on three sides and a fence costing $8 per foot on one side?
Answers
Answered by
GPT-4o mini
To solve this problem, we need to minimize the total cost of materials while maintaining a rectangular garden with a fixed area of 100 square feet. Let's denote the dimensions of the rectangle as follows:
- Let \( x \) be the length of the garden (one of the sides with the fence).
- Let \( y \) be the width of the garden (the other side).
The area constraint is given by:
\[
A = x \cdot y = 100
\]
From this, we can express \( y \) in terms of \( x \):
\[
y = \frac{100}{x}
\]
Now, we consider the cost of materials. The cost structure is as follows:
1. A brick wall costing $15 per foot on three sides:
- Two sides of length \( y \) (the width),
- One side of length \( x \) (the length).
The cost for the brick wall can be expressed as:
\[
\text{Cost of brick wall} = 15(2y + x) = 15(2\frac{100}{x} + x)
\]
2. A fence costing $8 per foot on one side (which is the length \( x \)):
\[
\text{Cost of fence} = 8x
\]
The total cost \( C \) can then be expressed as:
\[
C = 15\left(2\frac{100}{x} + x\right) + 8x
\]
Now substituting \( y \):
\[
C = 15\left(\frac{200}{x} + x\right) + 8x
\]
Combine the terms:
\[
C = \frac{3000}{x} + 15x + 8x
\]
\[
C = \frac{3000}{x} + 23x
\]
Now we will find the critical points of this cost function by taking the derivative and setting it to zero. First, we compute the derivative \( C' \):
\[
C' = -\frac{3000}{x^2} + 23
\]
Setting the derivative equal to zero gives:
\[
-\frac{3000}{x^2} + 23 = 0
\]
\[
\frac{3000}{x^2} = 23
\]
\[
3000 = 23x^2
\]
\[
x^2 = \frac{3000}{23}
\]
\[
x = \sqrt{\frac{3000}{23}} \approx 11.183
\]
Now we calculate \( y \) using the area equation:
\[
y = \frac{100}{x} = \frac{100}{\sqrt{\frac{3000}{23}}}
\]
\[
y = \sqrt{\frac{100^2 \cdot 23}{3000}} = \sqrt{\frac{2300}{30}} \approx 6.684
\]
Thus, the dimensions \( x \) and \( y \) that minimize the cost are approximately:
- Length \( x \approx 11.183 \) feet
- Width \( y \approx 6.684 \) feet
Finally, providing the answer as dimensions:
The dimensions of the rectangular garden that minimize the cost of materials are approximately \( x \approx 11.18 \) feet and \( y \approx 6.68 \) feet.
- Let \( x \) be the length of the garden (one of the sides with the fence).
- Let \( y \) be the width of the garden (the other side).
The area constraint is given by:
\[
A = x \cdot y = 100
\]
From this, we can express \( y \) in terms of \( x \):
\[
y = \frac{100}{x}
\]
Now, we consider the cost of materials. The cost structure is as follows:
1. A brick wall costing $15 per foot on three sides:
- Two sides of length \( y \) (the width),
- One side of length \( x \) (the length).
The cost for the brick wall can be expressed as:
\[
\text{Cost of brick wall} = 15(2y + x) = 15(2\frac{100}{x} + x)
\]
2. A fence costing $8 per foot on one side (which is the length \( x \)):
\[
\text{Cost of fence} = 8x
\]
The total cost \( C \) can then be expressed as:
\[
C = 15\left(2\frac{100}{x} + x\right) + 8x
\]
Now substituting \( y \):
\[
C = 15\left(\frac{200}{x} + x\right) + 8x
\]
Combine the terms:
\[
C = \frac{3000}{x} + 15x + 8x
\]
\[
C = \frac{3000}{x} + 23x
\]
Now we will find the critical points of this cost function by taking the derivative and setting it to zero. First, we compute the derivative \( C' \):
\[
C' = -\frac{3000}{x^2} + 23
\]
Setting the derivative equal to zero gives:
\[
-\frac{3000}{x^2} + 23 = 0
\]
\[
\frac{3000}{x^2} = 23
\]
\[
3000 = 23x^2
\]
\[
x^2 = \frac{3000}{23}
\]
\[
x = \sqrt{\frac{3000}{23}} \approx 11.183
\]
Now we calculate \( y \) using the area equation:
\[
y = \frac{100}{x} = \frac{100}{\sqrt{\frac{3000}{23}}}
\]
\[
y = \sqrt{\frac{100^2 \cdot 23}{3000}} = \sqrt{\frac{2300}{30}} \approx 6.684
\]
Thus, the dimensions \( x \) and \( y \) that minimize the cost are approximately:
- Length \( x \approx 11.183 \) feet
- Width \( y \approx 6.684 \) feet
Finally, providing the answer as dimensions:
The dimensions of the rectangular garden that minimize the cost of materials are approximately \( x \approx 11.18 \) feet and \( y \approx 6.68 \) feet.