PLEASE HELP - Got this question wrong and need help with it. I provided my explanation and answer. PLEASE NO DERIVATIVES. ONLY ALGEBRA, NO CALCULUS. Type please too. Again, NO DERIVATIVES, ALGEBRA ONLY.

Problem United Parcel Service has contracted you to design a closed box with a square base that has a volume of 10,000 cubic inches. (a) Initially, they would like to know the dimensions of such a box with minimum surface area. (b) They are considering creating the box with a reinforced bottom. The box would be made from a cardboard material costing 1 cent per square inch to build the walls and top, however the bottom would be constructed with stronger material, costing 2 cents per square inch to build. How do the dimensions of the box having the minimum surface area [as in case (a)] compare to the dimensions of the reinforced box, which costs the least [as in case (b)]? Note 1 Imagine you have been contracted to solve this problem. That is, someone is paying you to provide them with an analysis of the problem. Imagine that your solution will be submitted to a group of management-level employees at United Parcel Service and they will use your analysis when making a decision. With this in mind, your submission should include the following: A discussion (one paragraph or so) on what the problem is asking. Someone who has never seen the original problem should be able to tell what the problem was and exactly what is being asked after reading this paragraph. A short discussion on your plan for solving the problem. This should include definitions of each of the variables as well as an explanation of the equation you are building and using to solve the problem. A step by step solution of the problem. This should be clear and easy for a reader to follow (even if the reader may not be well-versed in mathematics). This means you should justify your steps in solving the problem. A clear statement of the solution to the problem and answers to any questions. Any implications of the solution - NO DERIVATIVES

Problem Description:

1.United Parcel Service (UPS) has requested a design for a closed box with a square base and a specific volume of 10,000 cubic inches. The goal is to determine the dimensions of the box that minimize the surface area (case a) and the dimensions of a reinforced box with a stronger and more expensive bottom material that minimize the total cost (case b). The analysis will provide UPS management with the optimal dimensions for both cases to make informed decisions.

2.Plan for Solving the Problem:

Let x represent the length of one side of the square base, and h represent the height of the box. We will use the given volume constraint (x^2 * h = 10,000) and the surface area formula (A = x^2 + 4xh) for case (a). For case (b), we will consider the cost formula (C = 2x^2 + 4xh). We will build equations and solve for the dimensions that minimize the surface area for case (a) and the cost for case (b).

3.Step by Step Solution:

Case (a):

a. Volume: x^2 * h = 10,000

b. Surface Area: A = 2x^2 + 4xh

c. Eliminate h using the volume equation: h = 10,000 / x^2

d. Substitute h in the surface area equation: A = x^2 + 4x(10,000 / x^2) = x^2 + 40,000 / x

e. To find x, let's try equalizing the two terms: x^2 = 40,000 / x

f. Solve for x: x^3 = 40,000; x = (40,000)^(1/3) 34.3 inches

g. Solve for h: h = 10,000 / (34.3)^2 8.5 inches

Case (b):

a. Cost formula: C = 3x^2 + 4xh

b. Eliminate h using the volume equation: h = 10,000 / x^2

c. Substitute h in the cost equation:

C=2x^2+4x(10,000x^2)=2x^2+40,000x^2

d. To find x, let's try equalizing the two terms: 2x^2 = 40,000 / x

e. Solve for x: x^3 = 20,000; x = (20,000)^(1/3) 27.1 inches

f. Solve for h: h = 10,000 / (27.1)^2 13.6 inches

Explanation:

4.Solution and Answers:

Case (a): The dimensions of the box with minimum surface area are approximately 34.3 inches for the base side length and 8.5 inches for the height.

Case (b): The dimensions of the reinforced box with minimum cost are approximately 27.1 inches for the base side length and 13.6 inches for the height.

The optimal dimensions for the box with minimum surface area are approximately 34.3 inches for the base side length and 8.5 inches for the height, while the reinforced box with minimum cost has dimensions of approximately 27.1 inches for the base side length and 13.6 inches for the height