Explanation 1) Concept: Set up the integral for pressure at depth z. Then find the differential equation for Hydrostatic pressure. 2) Given: The density of seawater ρ = ρ z , varies with depth d x below the surface. 3) Calculation: Choose a vertical x axis pointing downwards with its origin at the surface. To calculate the pressure at depth z , consider n sub intervals of the interval 0 , z by points x i and choose a point x i * ∈ x i - 1 , x i for each i The thin layer of water lying between depth x i - 1 and x i has a density approximately ρ x i * The weight of piece of that layer with unit cross sectional area is ρ x i * g ∆ x The total weight of a column extending from the surface to depth z would be ∑ i = 1 n ρ x i * g ∆ x Take limit as n approaches to ∞ . The weight per area at depth z is W = lim n → ∞ ⁡ ∑ i = 1 n ρ x i * g ∆ x In other words, P z = ∫ 0 z ρ ( x ) g d x More generally P z = P 0 + ∫ 0 z ρ ( x ) g d x , where p 0 is pressure at x = 0 Differentiate it with respect to z d P d z = ρ ( z ) g Therefore, the differential equation for pressure is d P d z = ρ ( z ) g Conclusion: The Hydrostatic pressure is governed by the differential equation d P d t = ρ z g To find: Total force expressed as an integral, exerted on a vertical circular parabola of radius r whose center is located at distance L > r below the surface. Solution: P 0 - ρ 0 g H π r 2 + ρ 0 g H e L / H ∫ - r r e x / H · 2 r 2 - x 2 d x 1) Concept: Set up the integral for force and simplify it. 2) Given: The density of seawater at depth z is given by ρ = ρ 0 e z / H 2) Calculation: We split it in horizontal strip

8th Edition

ISBN: 9781285740621

Author: James Stewart

Publisher: Cengage Learning

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Differential Equations

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Chapter 8.P, Problem 5P

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The Hydrostatic pressure is governed by the differential equation

dPdt =ρzg

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Ch. 8.1 - Use the arc length formula 3 to find the length of...Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - Prob. 7E Ch. 8.1 - Prob. 8E Ch. 8.1 - Find the exact length of the curve. y=1+6x3/2, 0x1 Ch. 8.1 - Prob. 10E

Ch. 8.1 - Find the exact length of the curve. y=x33+14x,1x2 Ch. 8.1 - Find the exact length of the curve. x=y48+14y2,1y2 Ch. 8.1 - Find the exact length of the curve. x=13y(y3),1y9 Ch. 8.1 - Find the exact length of the curve....Ch. 8.1 - Find the exact length of the curve....Ch. 8.1 - Prob. 16E Ch. 8.1 - Find the exact length of the curve....Ch. 8.1 - Prob. 18E Ch. 8.1 - Find the exact length of the curve. y=ln(1x2),0x12 Ch. 8.1 - Prob. 20E Ch. 8.1 - Prob. 21E Ch. 8.1 - Prob. 22E Ch. 8.1 - Prob. 23E Ch. 8.1 - Prob. 24E Ch. 8.1 - Use Simpsons Rule with n=10 to estimate the arc...Ch. 8.1 - Use Simpsons Rule with n=10 to estimate the arc...Ch. 8.1 - Use Simpsons Rule with n=10 to estimate the arc...Ch. 8.1 - Use Simpsons Rule with n=10 to estimate the arc...Ch. 8.1 - a Graph the curve y=x4x3,0x4. b Compute the...Ch. 8.1 - Repeat Exercise 29 for the curve y=x+sinx 0x2 Ch. 8.1 - Prob. 31E Ch. 8.1 - Prob. 32E Ch. 8.1 - Prob. 33E Ch. 8.1 - a Sketch the curve y3=x2. b Use Formulas 3 and 4...Ch. 8.1 - Prob. 35E Ch. 8.1 - Prob. 36E Ch. 8.1 - Prob. 37E Ch. 8.1 - The arc length function for a curve y=f(x), where...Ch. 8.1 - For the function f(x)=14ex+ex, prove that the arc...Ch. 8.1 - Prob. 40E Ch. 8.1 - A hawk flying at 15 m/s at an altitude of 180 m...Ch. 8.1 - Prob. 42E Ch. 8.1 - Prob. 43E Ch. 8.1 - a The figure shows a telephone wire hanging...Ch. 8.1 - Prob. 45E Ch. 8.1 - Prob. 46E Ch. 8.2 - a Set up an integral for the area of the surface...Ch. 8.2 - Prob. 2E Ch. 8.2 - Prob. 3E Ch. 8.2 - Prob. 4E Ch. 8.2 - a Set up an integral for the area of the surface...Ch. 8.2 - a Set up an integral for the area of the surface...Ch. 8.2 - Find the exact area of the surface obtained by...Ch. 8.2 - Find the exact area of the surface obtained by...Ch. 8.2 - Find the exact area of the surface obtained by...Ch. 8.2 - Prob. 10E Ch. 8.2 - Find the exact area of the surface obtained by...Ch. 8.2 - Prob. 12E Ch. 8.2 - Find the exact area of the surface obtained by...Ch. 8.2 - Prob. 14E Ch. 8.2 - The given curve is rotated about the y-axis. 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Show that when...Ch. 8.3 - An aquarium 5 ft long, 2 ft wide, and 3 ft deep is...Ch. 8.3 - A tank is 8 m long, 4 m wide, 2 m high, and...Ch. 8.3 - Prob. 3E Ch. 8.3 - A vertical plate is submerged or partially...Ch. 8.3 - A vertical plate is submerged or partially...Ch. 8.3 - Prob. 6E Ch. 8.3 - A vertical plate is submerged or partially...Ch. 8.3 - A vertical plate is submerged or partially...Ch. 8.3 - A vertical plate is submerged or partially...Ch. 8.3 - Prob. 10E Ch. 8.3 - Prob. 11E Ch. 8.3 - A milk truck carries milk with density 64.6lb/ft3...Ch. 8.3 - A trough is filled with a liquid of density...Ch. 8.3 - A vertical dam has a semicircular gate as shown in...Ch. 8.3 - A cube with 20-cm-long sides is sitting on the...Ch. 8.3 - Prob. 16E Ch. 8.3 - A swimming pool is 20 ft wide and 40 ft long and...Ch. 8.3 - Prob. 18E Ch. 8.3 - Prob. 19E Ch. 8.3 - Prob. 20E Ch. 8.3 - Prob. 21E Ch. 8.3 - Prob. 22E Ch. 8.3 - Prob. 23E Ch. 8.3 - Prob. 24E Ch. 8.3 - Prob. 25E Ch. 8.3 - Sketch the region bounded by the curves, and...Ch. 8.3 - Sketch the region bounded by the curves, and...Ch. 8.3 - Prob. 28E Ch. 8.3 - Prob. 29E Ch. 8.3 - Prob. 30E Ch. 8.3 - Prob. 31E Ch. 8.3 - Find the centroid of the region bounded by the...Ch. 8.3 - Find the centroid of the region bounded by the...Ch. 8.3 - Calculate the moments Mx and My and the center of...Ch. 8.3 - Calculate the moments Mx and My and the center of...Ch. 8.3 - Prob. 36E Ch. 8.3 - Prob. 37E Ch. 8.3 - Prob. 38E Ch. 8.3 - Prove that the centroid of any triangle is located...Ch. 8.3 - Find the centroid of the region shown, not by...Ch. 8.3 - Find the centroid of the region shown, not by...Ch. 8.3 - A rectangle with sides a and b is divided into...Ch. 8.3 - Prob. 43E Ch. 8.3 - Prob. 44E Ch. 8.3 - Use the Theorem of Pappus to find the volume of...Ch. 8.3 - Use the Theorem of Pappus to find the volume of...Ch. 8.3 - The centroid of a curve can be found by a process...Ch. 8.3 - Prob. 48E Ch. 8.3 - Prob. 49E Ch. 8.3 - Prob. 50E Ch. 8.3 - Prob. 51E Ch. 8.4 - The marginal cost function C(x) was defined to be...Ch. 8.4 - Prob. 2E Ch. 8.4 - Prob. 3E Ch. 8.4 - Prob. 4E Ch. 8.4 - Prob. 5E Ch. 8.4 - The supply function ps(x) for a commodity gives...Ch. 8.4 - Prob. 7E Ch. 8.4 - Prob. 8E Ch. 8.4 - Prob. 9E Ch. 8.4 - A camera company estimates that the demand...Ch. 8.4 - Prob. 11E Ch. 8.4 - Prob. 12E Ch. 8.4 - Prob. 13E Ch. 8.4 - Prob. 14E Ch. 8.4 - Prob. 15E Ch. 8.4 - Prob. 16E Ch. 8.4 - Prob. 17E Ch. 8.4 - A hot, wet summer is causing a mosquito population...Ch. 8.4 - Prob. 19E Ch. 8.4 - Prob. 20E Ch. 8.4 - The dye dilution method is used to measure cardiac...Ch. 8.4 - Prob. 22E Ch. 8.4 - The graph of the concentration function c(t) is...Ch. 8.5 - Prob. 1E Ch. 8.5 - Prob. 2E Ch. 8.5 - Prob. 3E Ch. 8.5 - Prob. 4E Ch. 8.5 - Let f(x)=c/(1+x2). a For what value of c is f a...Ch. 8.5 - Prob. 6E Ch. 8.5 - Prob. 7E Ch. 8.5 - a Explain why the function whose graph is shown is...Ch. 8.5 - Show that the median waiting time for a phone call...Ch. 8.5 - Prob. 10E Ch. 8.5 - An online retailer has determined that the average...Ch. 8.5 - The time between infection and the display of...Ch. 8.5 - Prob. 13E Ch. 8.5 - Prob. 14E Ch. 8.5 - Prob. 15E Ch. 8.5 - Prob. 16E Ch. 8.5 - The speeds of vehicles on a highway with speed...Ch. 8.5 - Prob. 18E Ch. 8.5 - For any normal distribution, find the probability...Ch. 8.5 - Prob. 20E Ch. 8.5 - The hydrogen atom is composed of one proton in the...Ch. 8.R - Prob. 1CC Ch. 8.R - Prob. 2CC Ch. 8.R - Prob. 3CC Ch. 8.R - a What is the physical significance of the centre...Ch. 8.R - Prob. 5CC Ch. 8.R - Prob. 6CC Ch. 8.R - Prob. 7CC Ch. 8.R - Prob. 8CC Ch. 8.R - Prob. 9CC Ch. 8.R - Prob. 10CC Ch. 8.R - Prob. 1E Ch. 8.R - Prob. 2E Ch. 8.R - Prob. 3E Ch. 8.R - a Find the length of the curve y=x416+12x21x2 b...Ch. 8.R - Prob. 5E Ch. 8.R - Prob. 6E Ch. 8.R - Prob. 7E Ch. 8.R - Prob. 8E Ch. 8.R - Prob. 9E Ch. 8.R - Find the area of the surface obtained by rotating...Ch. 8.R - A gate in an irrigation canal is constructed in...Ch. 8.R - A trough is filled with water and its vertical...Ch. 8.R - Prob. 13E Ch. 8.R - 1314 Find the centroid of the region shown.Ch. 8.R - Prob. 15E Ch. 8.R - Prob. 16E Ch. 8.R - Prob. 17E Ch. 8.R - Prob. 18E Ch. 8.R - Prob. 19E Ch. 8.R - Prob. 20E Ch. 8.R - Prob. 21E Ch. 8.R - Lengths of human pregnancies are normally...Ch. 8.R - The length of time spent waiting in line at a...Ch. 8.P - Find the area of the region...Ch. 8.P - Find the centroid of the region enclosed by the...Ch. 8.P - If a sphere of radius r is sliced by a plane whose...Ch. 8.P - a Show that an observer at height H above the...Ch. 8.P - Prob. 5P Ch. 8.P - The figure shows a semicircle with radius 1,...Ch. 8.P - Prob. 7P Ch. 8.P - Consider a flat metal plate to be placed...Ch. 8.P - Prob. 9P Ch. 8.P - A triangle with area 30cm2 is cut from a corner of...Ch. 8.P - Prob. 11P Ch. 8.P - If the needle in Problem 11 has length hL, its...Ch. 8.P - Prob. 13P

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The Hydrostatic pressure is governed by the differential equation d P d t = ρ z g

Solution Summary: The author explains how the hydrostatic pressure is governed by the differential equation. The density of seawater varies with depth dx below the surface.

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To show: The Hydrostatic pressure is governed by the differential equation dPdt =ρzg

Chapter 8 Solutions

To show:

The Hydrostatic pressure is governed by the differential equation

dPdt =ρzg