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A rectangular plate 10ft long and 6ft wide is submerged vertically with the shorter side 2ft below the water surface. Find the force on one side of the plate
Find the volume of the solid generated by revolving the region bounded by the curves x=y^2, x=2, y=0; about y=0
Work
Problem 5: You are in charge of the evacuation and repair of the storage tank shown in the figure below. The tank is a hemisphere of radius 10 ft and is full of benzene weighing 56 lb / ft^3 . A firm you contacted says it can empty the tank for 1/2 Β’ per foot-pound of work. Find the work required to empty the tank by pumping the benzene to an outlet 2 ft above the top of the tank. If you have $5,000 budgeted for the job, can you afford to hire the firm?Β
WorkΒ
Problem 1: A force of 500 dynes stretches a spring from its natural length of 20 cm to a length of 24 cm. Find the work done in stretching the spring from its natural length to a length of 28 cm.
Problem 2: A spring has a natural length of 6 in. A 1200-lb force compresses it to 5 1/2 in. Find the work done in compressing it from 6 in to 4 1/2 in.
Problem 3: An upright right-circular cylindrical tank of radius 5 ft and height 10 ft is filled with water. (a) How much work is done by pumping the water to the top of the tank? (b) Find the work required to pump the water to a level of 4 ft above the top of the tank.
Problem 4: A conical reservoir 10 m deep and 8m across the top is filled with water to a depth of 5 m. The reservoir is emptied by pumping the water over the top edge. How much work is done in the process?
Force due to Liquid PressureΒ
Problem 1: A horizontal cylindrical pipe has a 4-ft inner diameter and is closed at one end by a circular gate that just fits over the pipe. If the pipe contains water at a depth of 3 ft, find the force on the gate due to water pressure.
Problem 2: A gate is in the form of a parabola π₯^2 = 8π¦. Find the force on one side if it is submerged vertically with the focal chord (latus rectum) along the surface of water.
Problem 3: A gate is in the form of letter T with the vertical 2 ft wide and 4 ft high. The horizontal crosspiece is 6 ft long and 1 ft high. Find the force on one side if the gate is submerged vertically with the top of the crosspiece along the surface of water.
Problem 4: An automobileβs gasoline tank is in the shape of a right-circular cylinder of radius 8 in with a horizontal axis. Find the force on one end when the gasoline is 12 in deep if 0.39 oz / in^3 is the weight density of the gasoline.Β
Area of Surface of Revolution
1. 9π¦ = π₯^2 , 0 β€ π₯ β€ 2 ; x-axis
2. π¦ = ππ(π₯^2 β 1) , 2 β€ π₯ β€ 3 ; y-axis
3. π¦ = 3^βπ₯ , 0 β€ π₯ β€ 1 ; y-axis
4. π¦ = β9 β π₯^2 , β2 β€ π₯ β€ 2 ; x-axis
Length of ArcΒ
1. Find the length of arc of the curve π¦ = (π₯/2)^3/2 from the point where π₯ = 0 to the point where π₯ = 2 . 2. Find the length of arc of the curve π₯ = π¦^4/4 + 1/8y^2 from the point where π¦ = 1 to the point where π¦ = 2.
3. Find the length of arc of the curve π¦ = ππ(π πππ₯) from π₯ = 0 to π₯ = π/4 .
4. Find the length of arc of the curve π₯^2/3 + π¦^2/3 = 1 in the first quadrant from the point where π₯ = 1/8 to the point where π₯ = 1.
Differentiate, y=ln(xlnx)
Consider the motion of a particle in one
dimension (x-axis) under the simultaneous influence of two forces. One force repels the particle from the location x=1, while the second one attracts the particle towards the location x = -1. In both cases, the force is inversely proportional to the distance from x = 1 and x = -1 respectively, but the proportionality constants may be different. Describe the trajectory of the particle over time, for different values of the two proportionality constants. Does the motion of the particle change qualitatively as you vary the proportionality constants?
β’ Define variables that capture the important components of the situation.
β’ Set up equations to describe the desired relationship between the variables you have defined.
β’ Describe how to solve the equations you set up.
β’ Interpret the solutions you obtained in the context of the original situation.
You have a piece of wire of 10 cm. You use the wire to form either a square or a
circle, or you cut the wire and form a square and a circle .If you cut and use a wire
of length a to form the square and the wire of length 10 - a to form the circle,
the area of the square and/or circle is given by the function
f(a) = (1/16) *a^2+ (1/4pi) (10 - a)^2 = 0.142*a^2 - 1. 592*a + 7.958, 0 <= a <= 10.
For what value of a will the area be a maximum?
2. The function P = f(t) = 300 - t square root of (100 - 2t) gives the weight (in mg) of a population
of bacteria t hours after the start of an experiment.
The domain of the function is [0, 40]
2.1 Find the critical points of the function if f prime of (t) = 3*t - 100/
square root of (100 - 2*t)
.
2.2 Use a number line to find and describe the local extremes of the function.
