Section 7.5 Scientific and Statistical Applications

1. Section 7.5 Scientific and Statistical Applications

2. a Find k: b How much work would be needed to stretch the spring 3m beyond its natural length? Review: Hooke’s Law: A spring has a natural length of 1m. A force of 24N stretches the spring to 1.8 m.

3. Remember when work is constant, Work = Force ● Displacement Remember when work is not constant, Work =

4. Book Ex. 2 A leaky 5 lb bucket is raised 20 feet The rope weighs 0.08 lb/ft. The bucket starts with 2 gal (16 lb) of water and is empty when it just reaches the top. Work: Bucket (constant): Water: The force is proportional to remaining rope.

5. A leaky 5 lb bucket is raised 20 feet The rope weighs 0.08 lb/ft. The bucket starts with 2 gal (16 lb) of water and is empty when it just reaches the top. Work: Bucket: Water:

6. A leaky 5 lb bucket is raised 20 feet The rope weighs 0.08 lb/ft. The bucket starts with 2 gal (16 lb) of water and is empty when it just reaches the top. Work: Bucket: Water: Rope: Total:

7. 4 ft 0 10 ft dx 10 5 ft Like Ex. 3 I want to pump the water out of this tank. How much work is done? 5 ft 4 ft The force is the weight of the water. The water at the bottom of the tank must be moved further than the water at the top. 10 ft Consider the work to move one “slab” of water:

8. 4 ft 0 10 ft dx 10 5 ft I want to pump the water out of this tank. How much work is done? 5 ft 4 ft 10 ft distance force

9. 4 ft 0 A 1 horsepower pump, rated at 550 ft-lb/sec, could empty the tank in just under 14 minutes! 10 ft dx 10 5 ft I want to pump the water out of this tank. How much work is done? 5 ft 4 ft 10 ft distance force

10. Example 3 (book) 2 ft 10 ft A conical tank is filled to within 2 ft of the top with salad oil weighing 57 lb/ft3. How much work is required to pump the oil to the rim? 10 ft Consider one slice (slab) first:

11. Example 3 (book) 2 ft 10 ft A conical tank if filled to within 2 ft of the top with salad oil weighing 57 lb/ft3. How much work is required to pump the oil to the rim? 10 ft

12. Example 3 (book) 2 ft 10 ft A conical tank if filled to within 2 ft of the top with salad oil weighing 57 lb/ft3. How much work is required to pump the oil to the rim? 10 ft

13. What is the force on the bottom of the aquarium? 2 ft 3 ft 1 ft

14. It is just a coincidence that this matches the first answer! 0 2 ft 2 3 ft What is the force on the front face of the aquarium? 2 ft Depth (and pressure) are not constant. If we consider a very thin horizontal strip, the depth doesn’t change much, and neither does the pressure. 3 ft 1 ft depth density area

15. We could have put the origin at the surface, but the math was easier this way. A flat plate is submerged vertically as shown. (It is a window in the shark pool at the city aquarium.) 2 ft Find the force on one side of the plate. 3 ft Depth of strip: Length of strip: Area of strip: 6 ft density depth area

16. Examples: The mean (or ) is in the middle of the curve. The shape of the curve is determined by the standard deviation . Normal Distribution: For many real-life events, a frequency distribution plot appears in the shape of a “normal curve”. 13.5% 34% 2.35% heights of 18 yr. old men 68% standardized test scores 95% lengths of pregnancies 99.7% time for corn to pop “68, 95, 99.7 rule”

17. Normal Probability Density Function: (Gaussian curve) Normal Distribution: The area under the curve from a to b represents the probability of an event occurring within that range. 13.5% 34% 2.35% “68, 95, 99.7 rule” If we know the equation of the curve we can use calculus to determine probabilities:

18. Normal Probability Density Function: (Gaussian curve) Normal Distribution: The good news is that you do not have to memorize this equation! In real life, statisticians rarely see this function. They use computer programs or graphing calculators with statistics software to draw the curve or predict the probabilities.