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FACULTY NOTES
The LTAs and Spinoffs are designed so that each professor can implement them in a way that is
consistent with his/her teaching style and course objectives. This may range from using the
materials as out-of-class projects with minimal in-class guidance to doing most of the work in
class. The LTAs and Spinoffs are amenable to small group cooperative work and typically
benefit from the use of some learning technology. Since the objective of the LTAs and Spinoffs
is to support the specific academic goals you have set for your students, the Faculty Notes are not
intended to be prescriptive. The purpose of the Faculty Notes is to provide information that
assists you to take full advantage of the LTAs and Spinoffs. This includes suggestions for
instruction as well as answers for the exercises.
FACULTY NOTES
LTA 2
NASA Aquatics Lab
Background Information
This LTA is written so that in Part III Exercise 2, students will develop a function for the load of
the form L ( x ) =
a x^2
, where x 2 is the radius of the pallet. In Part III Exercise 3, the students
should get a linear function of the height.
Mathematical topics covered include: Relations involving Measurement and Geometry Basic Level Rational Equations Basic Level Graphs, Geometry Rational Functions and Limits (intuitively).
Timeline: This Laboratory Technical Activity (LTA) is intended as a two hour laboratory activity where students interact with peers and instructors. It is suggested that students begin work on this project during one class period and take it home for further work either individually or collaboratively. They can then finish the project either outside of class or during the following class period.
Required Technology: Any graphing calculator or computer algebra system may be employed. The LTA assumes students have a basic knowledge of how to graph functions and perform elementary operations with the technology the instructor uses in class.
Level: Precalculus or College Algebra
Instructional Methodology: This LTA is intended for group activity either in class or take home or both. Instructors are better able to meet individual needs in this learning environment because the capable students may be able to proceed with little or no assistance.
Possible Modifications: The LTA could be easily modified to provide a more difficult exercise for students when constructing their initial load function. The first two sections could be replaced by a single paragraph which instructs the students to identify appropriate variables, constants, and units needed to solve the load problem.
One variation of this would be to have the students start the identification in small groups, share their results with a larger group, and successively larger groups until the entire class has discussed appropriate variables, and so forth. It might be possible to have the class split into two competitive groups to come up with the “best” list with more points awarded to those with the best list.
Another variation would be to have the students start on the identification of variables, etc., and then hand out Parts I and II after the students have worked on the problem for a while.
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Solutions
Note: Answers for LTA 2 have been rounded to the three most significant digits to be consistent with the accuracy of the data provided.
Part I - Relevant Quantities
Table 1 Quantity Constant or Variable Constant Value or Variable Symbol
Appropriate Units
number of gallons per cubic foot constant 7.48 (i) gallons/cubic foot volume of entire tank in cubic feet constant 75.4 (ii) cubic feet volume of entire tank in gallons constant 564 (iii) gallons
radius of the tank constant 2 feet
radius of the pallet variable r feet
height of the tank constant 6 feet
water level in the tank variable h feet
(i) Number of gallons per cubic foot =
lbs ft 3
lbs gal
lbs ft 3
gal lbs
gal ft 3
(ii) Volume of tank (in cubic feet) = V = π (2 ft ) 2 (6 ft ) = 75. 3982 ft 3
(iii) Volume of tank (in gallons) = V = (7.
gal ft 3 )(75.3982^ ft
3 ) = 564.13 gal
Part II - Relationships
- a) A 2. a) V 3. a) W
b) A = π r 2 b) V = 4 π h b) W = 784 h
- Weight of full tank of water = 8.
lbs gal
(564.13 gal ) = 4700 lbs (rounded to 3 signifiant digits)
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Part III - Modeling
- Distributed load of full tank =
4700 lbs π (2 ft )
2 =^374
lbs ft 2
- Independent variable: r (in feet) Dependent variable: L (in pounds per square foot)
Function: L =
π r^2
- Independent variable: h (in feet) Dependent variable: L (in pounds per square foot)
Function: L =
62.4 (4 π h ) 4 π
= 62.4 h
Part IV - Analysis
- The radius of the pallet can be no more than 6 feet because the distance from the center of
Tank A to the nearest obstruction is
inch on the floor plan.
3 4
inch ÷
inch per foot =
⋅ 8 feet = 6 feet
- Notice that in the following graph we have taken the domain to be the interval (0,6]. This will allow us to study the effect of making the radius of the pallet any number greater than 0 feet but less than or equal 6 feet.
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