A rectangular area of 3200 ft^2 is to be fenced off. Two opposite sides will use fencing costing

step 1 Now in this given question we assume the dimensions of the rectangle to be x and y. So we have b

A rectangular area of 3200 ft^2 is to be fenced off. Two...

A rectangular area of $3200 \mathrm{ft}^{2}$ is to be fenced off. Two
opposite sides will use fencing costing $\$ 1$ per foot and the
remaining sides will use fencing costing $\$ 2$ per foot. Find
the dimensions of the rectangle of least cost.

Key Concepts

Optimization Under Constraints

This concept involves finding the minimum or maximum value of a function while satisfying specific conditions. In problems like these, a geometric quantity (such as area) is fixed, and another quantity (such as cost) is optimized subject to that constraint.

Constraint Equations

A constraint equation represents a condition that must be met by the variables involved. In geometric problems, it often relates dimensions to a fixed area or perimeter. It allows one to express one variable in terms of others, reducing the number of independent variables.

Cost Function Modeling

Cost function modeling is the process of expressing the total cost in terms of the variables that describe the problem, taking into account different costs associated with different parts. This function serves as the objective to optimize, allowing the determination of dimensions that minimize or maximize the cost.

Substitution Method

The substitution method is used to reduce a problem with multiple variables into one with a single variable by replacing one variable using the constraint equation. This simplification is crucial for making the optimization process manageable with calculus.

Differentiation and Critical Points

Differentiation involves computing the derivative of the cost function with respect to the chosen variable, which indicates how the cost changes as that variable changes. Setting the derivative to zero helps identify critical points — potential candidates for the minimum or maximum cost.

Second Derivative Test

The second derivative test is a method used to determine whether a critical point is a minimum or a maximum. By evaluating the second derivative at that point, one can confirm if the function is concave up (minimum) or concave down (maximum), ensuring that the optimal dimensions indeed minimize the cost.

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