Question
What is the solution to a problem involving the cost of fencing a rectangular lot with different fencing costs for length and width?
Answers
GPT-4o mini
To solve a problem involving the cost of fencing a rectangular lot with different costs for the length and the width, we can follow these general steps:
1. **Define the Variables**:
- Let \( L \) be the length of the rectangular lot.
- Let \( W \) be the width of the rectangular lot.
- Let \( C_L \) be the cost per unit length of fencing for the length.
- Let \( C_W \) be the cost per unit length of fencing for the width.
2. **Calculate the Total Cost of Fencing**:
The total cost \( C \) for fencing the rectangular lot can be expressed as:
\[
C = (Cost \text{ of Length Fencing}) + (Cost \text{ of Width Fencing})
\]
The lengths of fencing can be counted as follows:
- There are two lengths, so the cost contribution from length is \( 2 \times L \times C_L \).
- There are two widths, so the cost contribution from width is \( 2 \times W \times C_W \).
Thus, the total cost equation is:
\[
C = 2L \cdot C_L + 2W \cdot C_W
\]
3. **Set Up the Constraint**:
If there are constraints on the area \( A \) of the lot (for example, if you need to maintain a certain area), that can be expressed as:
\[
A = L \cdot W
\]
You can express one variable in terms of the other:
\[
W = \frac{A}{L}
\]
4. **Substitute and Formulate**:
Substitute for \( W \) in the total cost equation, if necessary:
\[
C = 2L \cdot C_L + 2 \left(\frac{A}{L}\right) \cdot C_W
\]
Now you have an equation with one variable \( L \).
5. **Optimize the Cost**:
To find the optimal dimensions that minimize the cost, take the derivative of the cost function \( C \) with respect to \( L \) and set it to zero to find critical points. Then, determine whether those points represent a minimum by checking the second derivative or conducting a sign analysis.
6. **Final Solution**:
Solve for the dimensions \( L \) and \( W \) based on your findings and constraints. This could involve additional steps if you need to consider non-negativity constraints or particular conditions regarding the layout.
The specific details will depend on the cost values, area constraints, and any additional parameters provided in the original problem.
**Example**:
Suppose you have a rectangular lot with a fixed area of 100 square meters, the cost of fencing along the length is $5 per meter, and the cost along the width is $3 per meter. You would set up your equations based on these parameters and follow through the optimization steps outlined above to find \( L \) and \( W \).
1. **Define the Variables**:
- Let \( L \) be the length of the rectangular lot.
- Let \( W \) be the width of the rectangular lot.
- Let \( C_L \) be the cost per unit length of fencing for the length.
- Let \( C_W \) be the cost per unit length of fencing for the width.
2. **Calculate the Total Cost of Fencing**:
The total cost \( C \) for fencing the rectangular lot can be expressed as:
\[
C = (Cost \text{ of Length Fencing}) + (Cost \text{ of Width Fencing})
\]
The lengths of fencing can be counted as follows:
- There are two lengths, so the cost contribution from length is \( 2 \times L \times C_L \).
- There are two widths, so the cost contribution from width is \( 2 \times W \times C_W \).
Thus, the total cost equation is:
\[
C = 2L \cdot C_L + 2W \cdot C_W
\]
3. **Set Up the Constraint**:
If there are constraints on the area \( A \) of the lot (for example, if you need to maintain a certain area), that can be expressed as:
\[
A = L \cdot W
\]
You can express one variable in terms of the other:
\[
W = \frac{A}{L}
\]
4. **Substitute and Formulate**:
Substitute for \( W \) in the total cost equation, if necessary:
\[
C = 2L \cdot C_L + 2 \left(\frac{A}{L}\right) \cdot C_W
\]
Now you have an equation with one variable \( L \).
5. **Optimize the Cost**:
To find the optimal dimensions that minimize the cost, take the derivative of the cost function \( C \) with respect to \( L \) and set it to zero to find critical points. Then, determine whether those points represent a minimum by checking the second derivative or conducting a sign analysis.
6. **Final Solution**:
Solve for the dimensions \( L \) and \( W \) based on your findings and constraints. This could involve additional steps if you need to consider non-negativity constraints or particular conditions regarding the layout.
The specific details will depend on the cost values, area constraints, and any additional parameters provided in the original problem.
**Example**:
Suppose you have a rectangular lot with a fixed area of 100 square meters, the cost of fencing along the length is $5 per meter, and the cost along the width is $3 per meter. You would set up your equations based on these parameters and follow through the optimization steps outlined above to find \( L \) and \( W \).