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MACC.912.G-MG.1.1: Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). MACC.912.G-MG.1.3: Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios)

Math & Nature Geometry & Biology ◦ CCSS G.MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). G.MG.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). G.MG.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

Math & Nature Geometry & Biology ◦ CCSS MACC.912.G-MG.1.1 Use geometric shapes, their measures, and their properties to describe objects (e.g. modeling a tree trunk or a human torso as a cylinder.) MACC.912.G-SRT.2.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Math & Nature Geometry & Biology ◦ CCSS MACC.912.G-MG.1.1: Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). MACC.K12.MP.1.1: Make sense of problems and persevere in solving them. MACC.K12.MP.4.1: Model with mathematics

Standards Addressed In Lesson MACC.912.G-SRT.3.6: Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. MACC.912.G-SRT.3.7: Explain and use the relationship between the sine and cosine of complementary angles. MACC.912.G-SRT.3.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Low Back Biomechanics of Lifting (cont.) M =W xh+W xb load & torso load torso Where: h – horizontal distance from load to L5/S1 disk b – horizontal distance from center of mass of the torso to the L5/S1 disk M back-muscle = F back-muscle x 5(N–cm) S(moments at L5/S1 disk = 0) F back-muscle x 5 = W load x h + W torso x b F back-muscle = (W load x h + W torso x b)/5 Assume h = 40 cm & b = 20 cm then F back-muscle = 8W load + 4W torso Assume W load = 300 N or 30kg (75lb) & W torso = 350 N (80lb) then

Schreiber, Schwöbbermeyer [12] proposed flexible pattern finder (FPF) in a system Mavisto.[23] It exploits downward closure , applicable for frequency concepts F2 and F3. The downward closure property asserts that the frequency for sub-graphs decrease monotonically by increasing the size of sub-graphs; but it does not hold necessarily for frequency concept F1. FPF is based on a pattern tree (see figure) consisting of nodes that represents different graphs (or patterns), where the parent is a sub-graph of its children nodes; i.e., corresp. graph of each pattern tree’s node is expanded by adding a new edge of its parent node. At first, FPF enumerates and maintains info of all matches of a sub-graph at the root of the pattern tree. Then builds child nodes of previous node by adding 1 edge supported by a matching edge in target graph, tries to expand all of previous info about matches to the new sub-graph (child node).[In next step, it decides whether the frequency of the current pattern is lower than a predefined threshold or not. If it is lower and if downward closure holds, FPF can abandon that path and not traverse further in this part of the tree; as a result, unnecessary computation is avoided. This procedure is continued until there is no remaining path to traverse. It does not consider infrequent sub-graphs and tries to finish the enumeration process as soon as possible; therefore, it only spends time for promising nodes in the pattern tree and discards all other nodes. As an added bonus, the pattern tree notion permits FPF to be implemented and executed in a parallel manner since it is possible to traverse each path of the pattern tree independently. But, FPF is most useful for frequency concepts F2 and F3, because downward closure is not applicable to F1. Still the pattern tree is still practical for F1 if the algorithm runs in parallel. It has no limitation on motif size, which makes it more amenable to improvements. ESU (FANMOD) Sampling bias of Kashtan et al. [9] provided great impetus for designing better algs for NM discovery, Even after weighting scheme, this method imposed an undesired overhead on the running time as well a more complicated impl. It supports visual options and is time efficient. But it doesn’t allow searching for motifs of size 9. Wernicke [10] RAND-ESU is better than jfinder, based on the exact enumeration algorithm ESU, has been implemented as an app called FANMOD.[10] Rand-esu is a discovery alg applicable for both directed and undirected networks. It effectively exploits an unbiased node sampling, and prevents overcounting sub-graphs. RAND-ESU uses DIRECT for determining sub-graph significance instead of an ensemble of random networks as a Null-model. DIRECT estimates sub-graph # w/oexplicitly generating random networks.[10] Empirically, DIRECT is more efficient than random network ensemble for sub-graphs with a very low concentration. But classical Null-model is faster than DIRECT for highly concentrated sub-graphs.[3][10] ESU alg: We show how this exact algorithm can be modified efficiently to RAND-ESU that estimates sub-graphs concentrations. The algorithms ESU and RAND-ESU are fairly simple, and hence easy to implement. ESU first finds the set of all induced sub-graphs of size k, let Sk be this set. ESU can be implemented as a recursive function; the running of this function can be displayed as a tree-like structure of depth k, called the ESU-Tree (see figure). Each of the ESU-Tree nodes indicate the status of the recursive function that entails two consecutive sets SUB and EXT. SUB refers to nodes in the target network that are adjacent and establish a partial sub-graph of size |SUB|≤k. If |SUB|=k, alg has found induced complete sub-graph, Sk=SUB ∪Sk. If |SUB|v} graphs of size 3 in the target graph. call ExtendSubgraph({v}, VExtension, v) endfor Leaves: set S3 or all of size-3 induced sub-graphs of the target graph (a). ESUtree nodes incl 2 adjoining sets: adjacent ExtendSubgraph(VSubgraph, VExtension, v) nodes called SUB and EXT=all adjacent if |VSubG|=k output G[VSubG] return 1 SUB node and where their numerical While VExt≠∅ do Remove arbitrary vertex w from VExt labels > SUB nodes labels. EXT set is VExtension′←VExtension∪{u∈Nexcl(w,VSubgraph)|u>v} utilized by the alg to expand a SUB set call ExtendSubgraph(VSubgraph ∪ {w}, VExtension′, v) until it reaches a desired size placed at return lowest level of ESU-Tree (or its leaves).