math lab

Project 2: Designing a Window for Maximum Illumination

RULES: You may work individually or in pairs, and the teams must be declared on the sign-in sheet today. This is a major assignment for this course. All work must be by your own team. You may consult only the following people when completing this assignment:

Besides the people named above, you may not consult other people (including, but not limited to, other students in the class, friends, tutors, math lab, family, etc). However, you may consult any sources such as your notes, book, other books, and (reliable) materials found online. Violation of this policy is considered academic dishonesty, which includes penalty of failing the class, so do not risk that. If you are not sure of the rules, come discuss with your instructor. There are 60 points attainable for this project. It is due two weeks after it is assigned. (Two points will be deducted for each weekday that the project is late, but no papers accepted beyond the last day of classes.)

U.R. Pei , an architect, is currently designing a large window for an eccentric client. The window will be made of stained glass. Its basic shape is parabolic, and there will be an imbedded right triangle within the window. The window is 9 feet high, and its base is also 6 feet. The parabola part can be modeled by the equation

x

x

y

6

2

+

-

=

from

6

0

£

£

x

. The base of the triangle forms part of the base of the window within it so that the left corner of the triangle is at the left corner of the window (see figure on the second page). The triangle will be stained yellow (grey part in figure), while the rest will be stained (dark) green.

Exercise: Find the dimensions of the triangle that provides the most yellow light possible, i.e. the maximum area of the triangle possible. Include units in your final answer and leave it in exact form (no decimal approximations). Do your work on (a) separate sheet(s) of paper, but staple these sheets on top as a cover.

Hints and reminders:

1. Place the figure on the Cartesian plane such that the base of the window is on the x-axis (

6

0

£

£

x

), and the vertex (top) of the parabola is on the point (3,9), where the parabola has equation

x

x

y

6

2

+

-

=

. Thus the bottom right corner of the triangle has coordinates (x,0).

2. Express the area of the triangle in terms of x. This is the (area) function you are maximizing.

3. You may use Maple or calculators to help you, but you must use calculus to solve the problem (involving the derivative, etc.) and show all your work independent of instruments or programs used.

4. Make sure you argue using calculus why you get a maximum (as opposed to say, minimum), with the dimensions of the triangle you found.

5. Make sure you state both the height and length of base of triangle at the end (with units).

Below list any print or electronic resources that you used (other than the textbook for Calculus I). If you did not consult other print or electronic sources write “No other sources consulted.” Sign at the bottom. By signing, you are also declaring that you followed all the rules stipulated at the beginning when completing the assignment. (You will lose points if you do not have a statement and if you do not sign.)

Signed________________________________________ Date______________________

Signed _______________________________________ Date______________________

Stained yellow

Stained green

Stained green

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