Model #111a
Foster Manufacturing Company - 1504 Armstrong Drive - Plano, Texas 75074-6027 - (972) 424-3644
CROSS-SECTIONAL SOLIDS TEACHERS GUIDE
Each of these models is hand-made, so the representative equations given may not be exactly the same as the ones which you get from your own data analysis. Also, bear in mind that when you compare your results with those of your students, the equations and resultant volume and area calculations can differ somewhat because of measurement differences that can result even from the width of the pencil line used. For example, using #111d, the difference in volume calculations between a theoretical base measurement and the one traced with a 0.7mm pencil is on the order of 0.4 cubic inches, or about a 5% difference. If you ask your students to provide exact numerical answers, consider accepting the answers within a +/- error range.
The best approach for performing the data analysis is to place the model on graph paper ( a 0.2" grid works fine) with the section on the y-axis, and some convenient point on the x-axis, as shown in the drawings below. Trace the outline of the base onto the graph paper. From this tracing, enter (x,y) points for both curves into a graphing calculator. If you use the same x-values for both curves, you have to enter them only once. Have the calculator perform a curve fit on both of the curves. #111a-#111c are straight lines intersecting a parabola. #111d is the intersection of 2 parabolas. For the sections, #111a is a right isosceles triangle with height-base parallel to the y-axis; #111b and #111c are squares with height=base parallel to the y-axis; and #111d is a right isosceles triangle with height= 1/2 the base parallel to the y-axis. These sections may be "seen" by holding the model over an overhead projector glass (with section parallel to the glass) and projecting a shadow of the section.
Note: The section embedded in each model is representative of the infinite number of sections which make up the solid. For #111a, #111b, and #111c, the section is on the axis of symmetry of the parabola. For #111d, the axes of symmetry of the two parabolas are parallel, and the section is between, and parallel, to these two axes.
Once the equations have been established, several topics can be discussed: area between curves, volume, length of arc, and error between data points and the curve fit equation. Also, for algebra and pre-calculus students, the base area/volume of the solids may be approximated using a Riemann Sum where the students sum areas/volumes of finite thickness sections.
Model #111a
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Model #111b
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Model #111c
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Model #111d
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