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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.)
