📘 optimization

1. **Problem statement:** A canvas wind shelter has a back, two square sides, and a top. The total canvas area is $147 \frac{1}{2}$ square feet. We want to find the depth that maxi

1. **Problem Statement:** A 216 m² rectangular pea patch is to be enclosed by a fence and divided into two equal parts by another fence parallel to one of the sides. We need to fin

1. **State the problem:** A rancher has resources: a feedlot for 200 steers, 100 hectares of land for sheep or barley. Each steer needs 0.5 tonnes barley. Profit per steer (excludi

1. The problem is to find the optimal values of variables $P$, $B$, $H$, and slack variables in an optimization context, typically linear programming. 2. The general approach uses

1. **Stating the problem:** We want to find the optimal combination of Monocrystalline and Polycrystalline solar panels to maximize total power output, considering budget and roof

1. **Problem statement:** Optimize the total cost function $$C = 3x^2 + 5xy + 6y^2$$ subject to the constraint $$5x + 7y = 1952$$. 2. **Method:** Use Lagrange multipliers to solve

1. **Problem Statement:** We want to maximize the profit function $$h(x,y) = 2x + 3y$$ subject to the constraints:

1. **Problem statement:** Maximize the function $$f(x,y) = x^{\frac{1}{2}} y^{\frac{1}{3}}$$ subject to the constraint $$3x + 4y \leq 25$$. 2. **Method:** We use the method of Lagr

1. **Problem Statement:** We need to use the Lagrange multiplier method to find the quantities of three inputs that minimize the cost function subject to a given constraint. 2. **G

1. **Problem Statement:** We are given a production function $$Q = e^{L^{0.15} K^{0.2} M^{0.1}}$$ and a cost function $$C = L^2 + K^2 + M^2$$ with constraints:

1. **Problem 2.2:** Maximize $w = \alpha x_1 + x_2$ subject to constraints: $$3x_1 + x_2 \leq 9$$

1. Planteamos el primer problema: "La suma de un número positivo y el doble de un segundo número positivo es 200. Hallar los dos números tales que su producto sea máximo." 2. Defin

1. Let's start by stating the problem: Linear programming is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements

1. Let's start by stating the problem: Linear programming is a method to find the maximum or minimum value of a linear objective function, subject to a set of linear inequalities o

1. **Problem Statement:** We want to maximize the function $$f(x_1,x_2) = x_1^2 - x_1 x_2 + x_2^2 + 2x_1 + 4x_2 + 3$$ subject to $$-5 \leq x_1, x_2 \leq 5$$ using Particle Swarm Op

1. **Problem Statement:** Minimize the objective function subject to given constraints. Since the specific function and constraints are not provided, let's outline the general appr

1. **State the problem:** We need to find the cereal box dimensions that maximize the volume-to-surface-area ratio while meeting the constraints: - Volume between 3400 cm³ and 3425

1. **Nyatakan masalah:** Diberikan fungsi biaya bahan bakar tiga stasiun pembangkit thermal:

1. The problem is to find a subset of items from an Excel file whose total value sums to exactly 12121624.47, with the constraint that quantities can be reduced but not increased.

1. State the problem: Gru wants to fence a rectangular dog pen along his house, so no fence is needed on the house side. He needs to find the dimensions that minimize fencing cost

1. **State the problem:** A man is 12 miles south of a straight beach and needs to reach a point 20 miles east along the shore. He can travel by motorboat to some point on the beac