Final answer:
To find the dimensions (length, width, height) with the smallest possible surface area of a rectangular tank given a volume of 500 cubic feet, we can use the Lagrange method. Set up the Lagrange equation using the volume equation as the constraint and the surface area equation as the objective function. Solve the system of equations formed by setting the partial derivatives of the Lagrange equation equal to zero to find the values of L, W, H, and λ.
Step-by-step explanation:
To find the dimensions (length, width, height) with the smallest possible surface area of the rectangular tank given a volume of 500 cubic feet, we can use the Lagrange method. Let's assume the length of the tank is L, the width is W, and the height is H. The volume equation for a rectangular tank is V = LWH. We want to minimize the surface area, which consists of the top, bottom, and four sides of the tank. The surface area equation is A = 2LW + 2LH + 2WH. We can set up the Lagrange equation using the volume equation as the constraint and the surface area equation as the objective function. The Lagrange equation is: f(L, W, H) = A - λ(V - 500), where λ is the Lagrange multiplier. To find the dimensions with the smallest possible surface area, we need to solve the system of equations formed by setting the partial derivatives of the Lagrange equation with respect to L, W, H, and λ equal to zero. After solving the system of equations, we will get the values of L, W, H, and λ. Then, substitute the values of L, W, and H back into the surface area equation to calculate the smallest possible surface area.