Math & Nature Fish feeding mechanisms ◦ Suction feeding Goliath grouper Epinephelus itajara Questions What fluid velocity can the goliath grouper generate during suction feeding? How does suction feeding by the goliath grouper compare to other fish?

Math & Nature References ◦ Bishop, K.L., Wainwright, P.C., and Holzman, R. (2008). Anterior to posterior wave of buccal expansion in suction feeding fish is critical for optimizing fluid flow velocity profile. Journal of the Royal Society, Interface. 5:1309-1316. ◦ Ferry-Graham, L.A., Wainwright, P.C., and Bellwood, D.R. (2001). Prey capture in long-jawed butterflyfishes (Chaetodontidae): the functional basis of novel feeding habits. Journal of Experimental Marine Biology and Ecology. 256:167-184. ◦ Galileo Galilei, The Assayer, as translated by Stillman Drake (1957), Discoveries and Opinions of Galileo pp. 237 - 238. New York: Doubleday & Company. ◦ Gibb, A.C. and Ferry-Graham, L.A. (2005). Cranial movements during suction feeding in teleost fishes: Are they modified to enhance suction production? Zoology. 108(2): 141-153. ◦ Grubich, J.R. (2001). Prey Capture in Actinopterygian Fishes: A Review of Suction Feeding Motor Patterns with New Evidence from an Elopomorph Fish, Megalops atlanticus. Integrative and Comparative Biology. 41(6): 1258-1265. ◦ Holzman, R., Day, S.W., and Wainwright, P.C. (2007). Timing is everything: coordination of strike kinematics affects the force exerted by suction feeding fish on attached prey. Journal of Experimental Biology. 210: 3328-3336. ◦ Holzman, R., Day, S.W., Mehta, R.S., and Wainwright, P.C. (2008). Jaw protrusion enhances forces exerted on prey by suction feeding fishes. Journal of the Royal Society, Interface. 5(29): 1445-1457.

PLACEMENT HOW? Placement Placement Test Test Transfer Transfer Credits Credits SAT SAT or or ACT ACT Scores Scores High High School School AP AP Scores Scores (Mathematics) (Mathematics) WHAT TO DO? Take Take placement placement test test on on any any campus campus at at Testing Testing Center. Center. Accuplacer Accuplacer MATH MATH can can be be taken taken twice. twice. Students Students who who wish wish higher higher placement placement must must then then “appeal”. “appeal”. Appeal Appeal is is request to take a pencil-and-paper type test on material from the course they request to take a pencil-and-paper type test on material from the course they placed placed into. into. Bring Bring transcript transcript showing showing general general education education math math credits credits from from another another college college or or university university to to CCBC CCBC for for evaluation. evaluation. If If completed completed in in Maryland, Maryland, the the highest highest level level developmental math course, Intermediate Algebra can be used for placement. developmental math course, Intermediate Algebra can be used for placement. Placement Placement by by SAT SAT MATH MATH score score of of 500 500 or or higher higher (or (or by by ACT ACT MATH MATH score score 21 21 or or higher) places the student into an entry-level general education math course higher) places the student into an entry-level general education math course (MATH (MATH 111, 111, 125, 125, 131/2/3, 131/2/3, 135, 135, 163). 163). For For higher higher placement, placement, the the student student must must take take Accuplacer MATH. MATH. Accuplacer Students Students with with documentation documentation of of AP AP math math scores scores of of 3, 3, 4, 4, or or 5 5 from from high high school school can can be awarded college credit and placement according to this chart. be awarded college credit and placement according to this chart.

Math & Nature Suction feeding in the goliath grouper ◦ Given Volume of the goliath grouper feeding mechanism at rest (t0) and at maximum expansion (t1) Duration of the feeding event (t1 - t0) Area of the mouth opening at maximum expansion (t1) 4) Find the velocity of water flow into the mouth of

Core Math MATH 95 MATH 96 MATH 126 MATH 176 MATH 96 MATH 126 MATH 176 MATH 126E MATH 176 Recommendation: If you start in Math 95 or Math 96 take summer classes to stay on track! MATH 126 MATH 176 MATH 181 or Math 182 will meet the College of Business math requirement. MATH 176 Pg. 12 of your Advising Manual

Math & Nature Suction feeding in the goliath grouper ◦ Given Dimensions of cones A and B at rest (t0) 1) Find the volume of the goliath grouper feeding mechanism at rest (t0). b c a a e d

Math & Nature Suction feeding in the goliath grouper ◦ Given Dimensions of cones A and B at rest (t0) 1) Find the volume of the goliath grouper feeding mechanism at rest (t0). c b a a e d

Math & Nature Suction feeding in the goliath grouper ◦ Given Dimensions of cones A and B at maximum expansion (t1) 2) Find the volume of the goliath grouper feeding mechanism at maximum expansion (t1).

Math questions • Evaluate the following expressions: – – – – – – – Math.abs(-1.23) Math.pow(3, 2) Math.pow(10, -2) Math.sqrt(121.0) - Math.sqrt(256.0) Math.round(Math.PI) + Math.round(Math.E) Math.ceil(6.022) + Math.floor(15.9994) Math.abs(Math.min(-3, -5)) • Math.max and Math.min can be used to bound numbers. Consider an int variable named age. – What statement would replace negative ages with 0? – What statement would cap the maximum age to 40? 27

Math & Nature Suction feeding in the goliath grouper ◦ Given Dimensions of cone B at maximum expansion (t1) 3) Find the area of the goliath grouper mouth at maximum expansion (t1). A. Collins mouth

Topological Sort A topological sort of an acyclic directed graph orders the vertices so that if there is a path from vertex u to vertex v, then vertex v appears after vertex u MATH 120 One topological sort in the ordering. of the course prerequisite graph at left: MATH 150 MATH 224 ECE 282 CS 111 CS 150 ECE 381 CS 111, MATH MATH 152 120, CS 140, MATH 423 MATH 125, CS ECE 482 MATH 250 150, MATH 224, CS 234 CS 240 STAT 380 CS 234, CS 240, ECE 483 MATH 321 ECE 282, CS 312, CS 312 CS 321 CS 325, MATH CS 325 150, MATH 152, CS 482 STAT 380, CS 321, CS 340 MATH 250, MATH CS 382 CS 434 321, CS 314, CS 314 MATH 423, CS CS 330 340, CS 425, ECE CS 425 381, CS 434, ECE CS 447 482, CS 330, CS CS 423 382, CS 423, CS CS 499 CS 438 CS 456 438, CS 454, CS 447, CS 499, CS CS 454 482, CS 456, ECE MATH 125 CS 340 CS 140 Page 5

Math Study Skills Inventory Rate your achievement of the following statements by placing a 3 for almost always, 2 for sometimes, and 1 for almost never. If you have never even thought about doing what the statement says, put a 0. Selecting a math class _______ 1. I schedule my math class at a time when I am mentally sharp. _______ 2. When I register for a math class, I choose the best instructor for me. _______ 3. If I have a choice, I select a math class that meets three or four days a week instead of one or two. _______ 4. I schedule the next math class as soon as possible after I have completed the current course. _______ 5. I am sure that I have signed up for the correct level math course. Time and place for studying math _______ 6. I study math every day. _______ 7. I try to get my math homework immediately after math class. _______ 8. I have a specific time to study math. _______ 9. I have a specific place with few distractions to study math. ______ 10. I seek help with my math homework in the lab/tutoring center. ______ 11. I am careful to keep up to date with math homework. ______ 12. I study math at least 8 to 10 hours a week.

What is Natural Heat Convection? • It relates to heat transfer (transmission) in fluids in MOTION. The fluid motion occurs by natural means such as buoyancy. • The MOTION of fluid can be created by a fan or pump with noticeable velocity (called Forced convection), or by difference of densities created by temperature differences in fluid in its natural state without external force applied to the fluid to cause the motion of the fluid. • The latter case of heat flow in the fluid is called “NATURAL CONVECTION.” Since the fluid velocity associated with natural convection is relatively low, the heat transfer coefficient encountered in natural convection is also low Convective heat transfer in the present analysis is related to the Boundary Layer that exists between the submerged solids and the surrounding fluids: • Boundary layer exists between a solid and its submerged fluid in motion. • Examples of boundary layers exist between the wings of airfoil cross sections of aircrafts, and also at the interfaces of hot or cold solid surface submerged in surrounding fluids at different temperatures. • Boundary layers act as “barriers” in air flow in the case of aircraft wings, and as a thermal barrier in the case of solids submerged in fluids • In the case in which a solid is submerged in a “virtual standstill” fluid but is subject to difference in temperatures such as illustrated in the figure to your right. What is Illustrated in the figure are the solid arrows indicate a hot solid submerged in a colder fluid at temperature Tf. The arrows in dotted lines shows the case with reversed temperature variations. • Because of the boundary layer at the Interface of the solid and fluid acts like a thermal barrier, the solid surface temperature T(rs, t) ≠ Tf – the bulk fluid temperature!! 2

Example: If Robinson Crusoe fishes by hand, he can catch 20 fish each week. If he takes a week off to make a net, he can then catch 25 fish a week with the net until it wears out in 10 weeks. In order to avoid starving during the week that he is weaving the net, he can borrow 10 fish from Friday, on the condition that he pays back the 10 fish plus an extra 5 fish. The cost of the net is the 20 fish that he gave up by not fishing for a week plus the 5 extra fish paid to Friday, or 25 fish. The gross marginal productivity of the net (the total addition to productivity that it contributes) is (5 fish per week)•(10 weeks) = 50 fish. The net marginal productivity of the net (the total addition to productivity that it contributes, less its cost) is (50 fish) – (25 fish) = 25 fish.

Goliath Grouper Suction Feeding Cone A Length (mm) Area (mm2) a 34.9 N/A b 153.6 Cone B Length (mm) Area (mm2) a 34.9 Time 0 c 54.3 N/A d 6.4 e Volume of feeding mechanism before expansion (t0) Cone A Length (mm) Area (mm2) a N/A b Cone B Length (mm) Area (mm2) a Time 1 c d e Volume of feeding mechanism at maximum expansion (t1) Volume change during feeding event (mm3) Duration of feeding event (sec) Area of mouth at maximum expansion (t1) (mm2) Velocity of water flow into mouth (mm/sec) Volume (mm3) Volume (mm3) Volume (mm3) Volume (mm3) 0.132

Goliath Grouper Suction Feeding Cone A Length (mm) Area (mm2) a 34.9 N/A b 153.6 Cone B Length (mm) Area (mm2) a 34.9 Time 0 c 54.3 N/A d 6.4 e 12.2 Volume of feeding mechanism before expansion (t0) Cone A Length (mm) Area (mm2) a N/A b Cone B Length (mm) Area (mm2) a Time 1 c d e Volume of feeding mechanism at maximum expansion (t1) Volume change during feeding event (mm3) Duration of feeding event (sec) Area of mouth at maximum expansion (t1) (mm2) Velocity of water flow into mouth (mm/sec) Volume (mm3) 195916.8 Volume (mm3) 84289.7 180206.5 Volume (mm3) Volume (mm3) 0.132

Goliath Grouper Suction Feeding Cone A Length (mm) Area (mm2) a 34.9 N/A b 153.6 Cone B Length (mm) Area (mm2) a 34.9 Time 0 c 54.3 N/A d 6.4 e 12.2 Volume of feeding mechanism before expansion (t0) Cone A Length (mm) Area (mm2) a 39.5 N/A b 161.3 Cone B Length (mm) Area (mm2) a 39.5 Time 1 c 56.4 d 32.6 e Volume of feeding mechanism at maximum expansion (t1) Volume change during feeding event (mm3) Duration of feeding event (sec) Area of mouth at maximum expansion (t1) (mm2) Velocity of water flow into mouth (mm/sec) Volume (mm3) 195916.8 Volume (mm3) 84289.7 280206.5 Volume (mm3) Volume (mm3) 0.132

Goliath Grouper Suction Feeding Cone A Length (mm) Area (mm2) a 34.9 N/A b 153.6 Cone B Length (mm) Area (mm2) a 34.9 Time 0 c 54.3 N/A d 6.4 e 12.2 Volume of feeding mechanism before expansion (t0) Cone A Length (mm) Area (mm2) a 39.5 N/A b 161.3 Cone B Length (mm) Area (mm2) a 39.5 Time 1 c 56.4 d 32.6 e 266.5 Volume of feeding mechanism at maximum expansion (t1) Volume change during feeding event (mm3) Duration of feeding event (sec) Area of mouth at maximum expansion (t1) (mm2) Velocity of water flow into mouth (mm/sec) Volume (mm3) 195916.8 Volume (mm3) 84289.7 280206.5 Volume (mm3) 263547.1 Volume (mm3) 230974.7 494521.7 214315.3 0.132

Goliath Grouper Suction Feeding Cone A Length (mm) Area (mm2) a 34.9 N/A b 153.6 Cone B Length (mm) Area (mm2) a 34.9 Time 0 c 54.3 N/A d 6.4 e 12.2 Volume of feeding mechanism before expansion (t0) Cone A Length (mm) Area (mm2) a 39.5 N/A b 161.3 Cone B Length (mm) Area (mm2) a 39.5 Time 1 c 56.4 3338.8 d 32.6 e 266.5 Volume of feeding mechanism at maximum expansion (t1) Volume change during feeding event (mm3) Duration of feeding event (sec) Area of mouth at maximum expansion (t1) (mm2) Velocity of water flow into mouth (mm/sec) Volume (mm3) 195916.8 Volume (mm3) 84289.7 280206.5 Volume (mm3) 263547.1 Volume (mm3) 230974.7 494521.7 214315.3 0.132 3338.8

Goliath Grouper Suction Feeding Cone A Length (mm) Area (mm2) a 34.9 N/A b 153.6 Cone B Length (mm) Area (mm2) a 34.9 Time 0 c 54.3 N/A d 6.4 e 12.2 Volume of feeding mechanism before expansion (t0) Cone A Length (mm) Area (mm2) a 39.5 N/A b 161.3 Cone B Length (mm) Area (mm2) a 39.5 Time 1 c 56.4 3338.8 d 32.6 e 266.5 Volume of feeding mechanism at maximum expansion (t1) Volume change during feeding event (mm3) Duration of feeding event (sec) Area of mouth at maximum expansion (t1) (mm2) Velocity of water flow into mouth (mm/sec) Volume (mm3) 195916.8 Volume (mm3) 84289.7 280206.5 Volume (mm3) 263547.1 Volume (mm3) 230974.7 494521.7 214315.3 0.132 3338.8 486