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

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

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 Goliath grouper model ◦ Objective Determine the velocity of water flow into the mouth ◦ Procedure Determine the volume of components A and B at rest (t0) and at maximum expansion (t1) t0 = time at rest t1 = time at maximum expansion A B Determine the volume change during feeding A B

Snook Suction Feeding Cone A Length (mm) Area (mm2) a 2.1 N/A b 27.6 Cone B Length (mm) Area (mm2) a 2.1 Time 0 c 12.3 N/A d 1.8 e Volume of feeding mechanism before expansion (t0) Cone A Length (mm) Area (mm2) a 7.0 N/A b 28.9 Cone B Length (mm) Area (mm2) a 7.0 Time 1 c 12.3 d 5.9 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) 0.036

Snook Suction Feeding Cone A Length (mm) Area (mm2) a 2.1 N/A b 27.6 Cone B Length (mm) Area (mm2) a 2.1 Time 0 c 12.3 N/A d 1.8 e 73.8 Volume of feeding mechanism before expansion (t0) Cone A Length (mm) Area (mm2) a 7.0 N/A b 28.9 Cone B Length (mm) Area (mm2) a 7.0 Time 1 c 12.3 109.4 d 5.9 e 66.0 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) 127.5 Volume (mm3) 147.2 274.7 Volume (mm3) 1482.9 Volume (mm3) 1611.5 3094.4 2819.7 0.036 109.4 716

Longjaw Butterfly Fish Suction Feeding Cone A Length (mm) Area (mm2) a 5.0 N/A b 14.9 Cone B Length (mm) Area (mm2) a 5.0 Time 0 c 31.2 N/A d 1.1 e Volume of feeding mechanism before expansion (t0) Cone A Length (mm) Area (mm2) a 5.0 N/A b 14.9 Cone B Length (mm) Area (mm2) a 5.0 Time 1 c 31.6 d 1.1 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) 0.022

Longjaw Butterfly Fish Suction Feeding Cone A Length (mm) Area (mm2) a 5.0 N/A b 14.9 Cone B Length (mm) Area (mm2) a 5.0 Time 0 c 31.2 N/A d 1.1 e 8.8 Volume of feeding mechanism before expansion (t0) Cone A Length (mm) Area (mm2) a 5.0 N/A b 14.9 Cone B Length (mm) Area (mm2) a 5.0 Time 1 c 31.6 3.8 d 1.1 e 8.9 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) 390.1 Volume (mm3) 1036.0 1426.1 Volume (mm3) 390.1 Volume (mm3) 1049.3 1439.4 13.3 0.022 3.8 159

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

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 & 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 Goliath grouper model ◦ Objective Determine the velocity of water flow into the mouth ◦ Procedure Determine the area of the mouth at maximum expansion (t1) t1 = time at maximum expansion A A B B