MATH125: Unit 1 Individual Project Mathematical Modeling and Problem Solving All commonly used formulas for geometric objects are really mathematical models of the characteristics of physical objects. For example, the characteristic of the volume inside a common closed cardboard box can be modeled by the formula for the volume of a rectangular solid, V = L x W x H, where L = Length, W = Width, and H = Height of the box. A basketball, because it is a sphere, can be partially modeled by its distance from one side through the center to the other side, or diameter, by the diameter formula for a sphere, D = 2r. Complete only ONE of the following questions. 1. (Please review Chapter 9 in the College Math text for geometric objects and their properties.) For a familiar example, the perimeter and area formulas for a rectangle are mathematical models for distance around the rectangle (perimeter) and area enclosed by the sides, respectively; P = 2L + 2W and A = L x W. For another example, the volume of a rectangular box would be: V = L x W x H, where L = Length, W = Width, and H = Height. The surface area of a rectangular box would be: SA = 2(L x W) + 2(W x H) + 2(L x H). Your problem is to obtain (or make) a rectangular box with a top on it that has the smallest possible surface area and that a football and a basketball, both fully inflated, will just fit into at the same time. What could make a good model for this situation? Using Polya's technique for solving problems, describe and discuss the strategy, steps, and procedures you will use to solve this problem. Then, demonstrate that your solution is correct. 2. (Please review Chapter 9 in the College Math text for geometric objects and their properties; walls, windows, and ceilings are all rectangles.) The walls and ceiling in your bedroom need to be painted, and the painters' estimates to do the work are far too expensive. You decide that you will paint the bedroom yourself. The bedroom is 14 ft. 3 in. by 16 ft., and the ceiling is 8 ft. high. The color of paint you have selected covers 75 sq. ft. per gallon, and costs $33.50 per gallon. The ceiling will be painted with a bright white ceiling paint that costs $28.50 per gallon but only covers 50 sq. ft. per gallon. There is one window in the room, and it is 3 ft. 4 in. by 5 ft. and will not be painted. The inside of the bedroom door is to be painted the same color as the walls. Describe and discuss how you will use Polya's problem-solving techniques to determine how much it will cost to paint this room with two coats of paint (on both walls and ceiling). Then, using your solution strategy, determine how much it will actually cost to paint your bedroom. Assuming you can paint 100 sq. ft. per hour, what will be the work time needed to paint your bedroom? (Because different paint lots of the same color may appear slightly different colors, when painting a room, you should buy all of your paint at one time and intermix the paint from at least two different cans so that the walls will all be exactly the same color.)