WebAssign Lesson 3-3 Applications (Homework)

Transcription

1 WebAssign Lesson 3-3 Applications (Homework) Current Score : / 27 Due : Tuesday, July :00 AM MDT Jaimos Skriletz Math 175, section 31, Summer Instructor: Jaimos Skriletz 1. /3 points A rectangular plate that is 10 m wide and 20 m long is submerged under water such that its top is level with the waters surface as shown. Use ρ = 1000 kg/m 3 for the density of water and g = 9.8 m/s 2 for gravitational acceleration. 1. What is the force on a small strip of thickness dy a distance y from the surface of the water? df = 2. Set up the definite integral and find the total force on this plate. F = df =

2 2. /3 points A large vertical dam is the shape of a symmetric trapezoid has a height of 30 m, a width of 20 m at its base, and a width of 40 m at the top as shown. Find the total force on the face of this dam when the reservoir is full (assume the water level is at the very top of the dam). Use ρ = 1000 kg/m 3 for the density of water and g = 9.8 m/s 2 for gravitational acceleration. 1. What is the force on a small strip of thickness dy a distance y up from the center of the bottom of the dam? df = 2. Set up the definite integral and find the total hydrostatic force on this dam. (Be accurate to at least 10 5 Newtons.) F = df =

3 3. /3 points A large vertical dam is the shape of a parabola given by the equation y = x 2 /16 where x and y are measured in meters as shown. Find the total force on the face of this dam when the reservoir is full (Assume the water level is at the very top of the dam). Use ρ = 1000 kg/m 3 for the density of water and g = 9.8 m/s 2 for gravitational acceleration. 1. What is the force on a small strip of thickness dy a distance y up from the center of the bottom of the dam? df = 2. Set up the definite integral and find the total hydrostatic force on this dam. (Be accurate to at least 10 4 Newtons.) F = df =

4 4. /2 points Consider the region bounded below the curve y = 3cos π x 4, to the right of the y-axis, and above the x-axis (only use the first region formed by the wave). If the linear distances are measured in feet and the density of this region is ρ = 25 g/ft 2, find the moment about the y-axis of this region. Be accurate to the nearest two decimal places. Moment about y-axis = Mass Distance to y-axis. M y = 5. /2 points A population of foxes has radial density ρ(r) = 6e 0.05r foxes per square mile where r is the distance from the center of their population. What is the total number of foxes in a 10 mile radius from their center? Round answer to the nearest whole number. Population = foxes

5 6. /2 points A plate in the shape of an isosceles triangle with base 3 m and height 6 m is submerged vertically in a tank of water so that its vertex is located 3 m below the surface of the water. Calculate the total fluid force F on a side of the plate. (The density of water is ρ = 1000 kg/m 3 and the gravitational acceleration g is 9.8 m/s 2. ) Calculate the total force on the side of this plate. Be accurate to the nearest 10 3 Newtons. Hydrostatic Force = 7. /2 points A circular disc as a variable thickness such that its radial density is given by ρ(r) = 12sin(0.25r + 0.5) g/in 2 where r is the distance from the center. What is the total mass of this disc if its radius is 8 inches? Be accurate to at least two decimal digits. Mass =

6 8. /2 points A spherical water tank with a radius of 3 meters is built 2 meters above the ground as shown. Find the total potential energy of the water in a full tank. Potential Energy = Mass g Height Use ρ = 1000 kg/m 3 for the density of water and g = 9.8 m/s 2 for the acceleration due to gravity. (Hint: Define your variable from the center of the tank.) (Be accurate to the nearest 10 3.) Potential Energy =

7 9. /2 points Calculate the fluid force on a side of the plate in the figure below, submerged in fluid of mass density ρ = 675 kg/m 3. (Use g = 9.8 m/s 2 for the acceleration due to gravity.) Be accurate to the nearest Hydrostatic Force = 10. /2 points A swimming pool has a rectangular surface that is 9 meters wide and 18 meters long. The shallow end is 1 meter deep and slopes down to 3 meters deep at the other end as shown. Find the amount of work required to pump all the water out of this swimming pool though a hose 0.2 meters above the top of the pool. The density of water is ρ = 1000 kg/m 3, g = 9.8 m/s 2 and Work = Force Distance = Mass g Height (Be accurate to the nearest 10^4 joules.) Work =

8 11. /2 points A massive dam has a height of 185 m. Calculate the force on the slanted face of the dam, assuming that the dam is a trapezoid of base 2000 m and upper edge 4000 m, inclined at an angle of 55 to the horizontal. (The density of water is ρ = 10 3 kg/m 3 and the gravitational acceleration g is 9.8 m/s 2. ) Hint: The thickness of a little area element is ds since the dam is slanted. Be accurate to the nearest 10 9 Hydrostatic Force =

9 12. /2 points A hemispherical dome with a radius of 10 m is built 30 m under the surface of the water as shown. Find the total hydrostatic force on the surface of this dome. Be accurate to the nearest Hint: Recall the surface area of a circular ring with radius r and thickness ds is da = 2 π r ds. Hydrostatic Force =