MATH125: Unit 1 Individual Project Mathematical Modeling and Problem Solving Be sure to show ALL of your work details. Submit your ANSWER FORM in the Unit 1 IP Submissions area. All commonly used formulas for geometric objects are really mathematical models of the characteristics of physical objects. For example, a basketball, because it is a sphere, can be partially modeled by its distance from one side through the center (radius, r) and then to the other side by the diameter formula for a sphere, D = 2r. For a familiar two-dimensional variables L : Length and W : Width, the perimeter and area formulas for a rectangle are mathematical models for distance around the rectangle (perimeter, P) and the region enclosed by the sides (area, A), respectively: P = 2L + 2W and A = L x W. Along with another variable H: Height, a three-dimensional rectangular prism's volume and surface area can be measured. For example, the formulas for a common closed cardboard box's inside space (volume, V) and outside covering (surface area, SA) are respectively: V = L x W x H and SA = 2(L x W) + 2(W x H) + 2(L x H). For this IP assignment, choose ONLY ONE question to complete in its entirety. Follow Polya's principles to solve your chosen problem. Explain your interpretation of what the problem is about. Develop and write down a strategy for solving this problem; show the steps in the correct order for your attempted solution. Did your strategy actually solve the problem? How do you know? Suppose your solution did not solve the problem, what would be your next action? QUESTION 1 Your goal is to construct a rectangular box with a top on it that has the smallest possible surface area, in which a football and a basketball, both fully inflated, will just fit into at the same time. Below are the measurements of the football and basketball. 9 . 5 in c h e s 6 .5 i n c h e s 1 1 . 5 in c h e s 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. Show all step-by-step calculations, include the units of measurement and round your final answers to the nearest whole measurement unit. Finally, demonstrate that your solution is correct. In other words, explain why the box you have created is the smallest possible box. QUESTION 2 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. Below is the information to help you solve the problem: The bedroom is 14 feet 3 in. by 16 feet, and the ceiling is 8 feet high. The color of paint you have selected for the walls covers 75 square feet per gallon, and costs $33.50 per gallon. The inside of the bedroom door is to be painted the same color as the walls. 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. Two coats of paint will be applied to all painted surface. The room has one window, measuring 3 ft. 4 in. by 5 ft., which will not be painted. 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? NOTE: Show all step-by-step calculations, include the units of measurement and round your final answers to the nearest whole measurement unit