Question
What are the dimensions of a rectangular garden fence with the lowest cost, given a fixed perimeter of 24 meters and $5 per meter of fencing?
Answers
GPT-4o mini
To minimize the cost of a rectangular garden fence given a fixed perimeter of 24 meters and a fencing cost of $5 per meter, we start by defining the dimensions of the rectangle and then calculating the cost based on those dimensions.
Let the length of the rectangle be \( L \) meters and the width be \( W \) meters. The perimeter \( P \) of the rectangle can be expressed as:
\[
P = 2L + 2W
\]
Given that the perimeter is fixed at 24 meters, we have:
\[
2L + 2W = 24
\]
Dividing the entire equation by 2 simplifies it to:
\[
L + W = 12
\]
To express one variable in terms of the other, we can set:
\[
W = 12 - L
\]
Next, we want to find the area \( A \) of the rectangle to determine the dimensions that minimize the cost:
\[
A = L \times W = L(12 - L) = 12L - L^2
\]
The cost \( C \) of the fencing can be calculated based on the total length of fencing needed, which is equal to the perimeter:
\[
C = 5 \times P = 5 \times 24 = 120 \, \text{dollars}
\]
Surprisingly, the cost is constant at $120 because the perimeter is fixed. Therefore, our goal is to find the dimensions that give a certain type of optimization.
However, since we already know the perimeter is fixed and the cost does not change with different dimensions as long as the perimeter is 24 meters, we must focus on maximizing the area.
When we express the area:
\[
A = 12L - L^2
\]
This is a quadratic equation that opens downwards (since the coefficient of \( L^2 \) is negative). To find the maximum area, we can use the vertex formula for a parabola \( L = -\frac{b}{2a} \) where \( A = aL^2 + bL + c \):
In our equation, \( a = -1 \) and \( b = 12 \):
\[
L = -\frac{12}{2 \times -1} = 6
\]
Now substituting \( L = 6 \) back into the expression for \( W \):
\[
W = 12 - L = 12 - 6 = 6
\]
Thus, the dimensions with the lowest cost (constant as it is related to the perimeter) - but optimal for area - are:
\[
L = 6 \, \text{meters} \quad \text{and} \quad W = 6 \, \text{meters}
\]
Finally, the minimum cost remains:
\[
\text{Cost} = 120 \, \text{dollars}
\]
The optimal dimensions of the rectangular garden fence are \( 6 \) meters by \( 6 \) meters, forming a square, which minimizes the area enclosed while maintaining the fixed perimeter.
Let the length of the rectangle be \( L \) meters and the width be \( W \) meters. The perimeter \( P \) of the rectangle can be expressed as:
\[
P = 2L + 2W
\]
Given that the perimeter is fixed at 24 meters, we have:
\[
2L + 2W = 24
\]
Dividing the entire equation by 2 simplifies it to:
\[
L + W = 12
\]
To express one variable in terms of the other, we can set:
\[
W = 12 - L
\]
Next, we want to find the area \( A \) of the rectangle to determine the dimensions that minimize the cost:
\[
A = L \times W = L(12 - L) = 12L - L^2
\]
The cost \( C \) of the fencing can be calculated based on the total length of fencing needed, which is equal to the perimeter:
\[
C = 5 \times P = 5 \times 24 = 120 \, \text{dollars}
\]
Surprisingly, the cost is constant at $120 because the perimeter is fixed. Therefore, our goal is to find the dimensions that give a certain type of optimization.
However, since we already know the perimeter is fixed and the cost does not change with different dimensions as long as the perimeter is 24 meters, we must focus on maximizing the area.
When we express the area:
\[
A = 12L - L^2
\]
This is a quadratic equation that opens downwards (since the coefficient of \( L^2 \) is negative). To find the maximum area, we can use the vertex formula for a parabola \( L = -\frac{b}{2a} \) where \( A = aL^2 + bL + c \):
In our equation, \( a = -1 \) and \( b = 12 \):
\[
L = -\frac{12}{2 \times -1} = 6
\]
Now substituting \( L = 6 \) back into the expression for \( W \):
\[
W = 12 - L = 12 - 6 = 6
\]
Thus, the dimensions with the lowest cost (constant as it is related to the perimeter) - but optimal for area - are:
\[
L = 6 \, \text{meters} \quad \text{and} \quad W = 6 \, \text{meters}
\]
Finally, the minimum cost remains:
\[
\text{Cost} = 120 \, \text{dollars}
\]
The optimal dimensions of the rectangular garden fence are \( 6 \) meters by \( 6 \) meters, forming a square, which minimizes the area enclosed while maintaining the fixed perimeter.