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Chapter 3: Q30E (page 103)

Old Man River keeps moving to suppose the man in Problem 28 again enters the current atbut this time decides to swim so that his velocity vectoris always directed toward the west beach. Assume that the speedis a constant. Show that a mathematical model for the path of the swimmer in the river is now

Short Answer

Expert verified

Answer

The mathematical model for the path of the swimmer in the river is now

Step by step solution

01

Define the chain rule of a mathematical model

The chain rule is the procedure for determining the derivative of a composite function (e.g.,, etc.).

The composite function rule is another name for it. Only composite functions are subject to the chain rule.

02

Find the path of the swimmer in the river.

The directions of the velocitiesand.

We can have the vectors

From equations a and b,

03

Use chain rule


Now substitute in the equation in 3

The path of the swimmer

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Most popular questions from this chapter

(a) Suppose \({\bf{a = b = 1}}\) in the Gompertz differential equation (7). Since the DE is autonomous, use the phase portrait concept of Section \({\bf{2}}{\bf{.1}}\) to sketch representative solution curves corresponding to the cases \({{\bf{P}}_{\bf{0}}}{\bf{ > e}}\) and \({\bf{0 < }}{{\bf{P}}_{\bf{0}}}{\bf{ < e}}\).

(b) Suppose \({\bf{a = 1,b = - 1}}\) in (7). Use a new phase portrait to sketch representative solution curves corresponding to the cases \({{\bf{P}}_{\bf{0}}}{\bf{ > }}{{\bf{e}}^{{\bf{ - 1}}}}\) and \({\bf{0 < }}{{\bf{P}}_{\bf{0}}}{\bf{ < }}{{\bf{e}}^{{\bf{ - 1}}}}\).

(c) Find an explicit solution of (7) subject to \({\bf{P(0) = }}{{\bf{P}}_{\bf{0}}}\).

Question: When all the curves in a family \(G\left( {x,y,{c_1}} \right) = 0\)intersect orthogonally all the curves in another family\(H\left( {x,y,{c_2}} \right) = 0\), the families are said to be orthogonal trajectories of each other. See Figure 3.R.5. If \(dy/dx = f(x,y)\)is the differential equation of one family, then the differential equation for the orthogonal trajectories of this family is\(dy/dx = - 1/f(x,y)\). In Problems\(\;{\bf{15}} - {\bf{18}}\)find the differential equation of the given family by computing\(dy/dx\)and eliminating \({c_1}\)from this equation. Then find the orthogonal trajectories of the family. Use a graphing utility to graph both families on the same set of coordinate axes.

\(y = {c_1}{e^x}\)

Skydiving A skydiver is equipped with a stopwatch and an altimeter. As shown in Figure 3.2.7, he opens his parachuteseconds after exiting a plane flying at an altitude offeet and observes that his altitude isfeet. Assume that air resistance is proportional to the square of the instantaneous velocity, his initial velocity on leaving the plane is zero, and.

(a) Find the distance, measured from the plane, the skydiver has travelled during freefall in time.

[Hint: The constant of proportionalityin the model given in Problem 15 is not specified. Use the expression for terminal velocityobtained in part (b) of Problem 15 to eliminatefrom the IVP. Then eventually solve for.

(b) How far does the skydiver fall and what is his velocity at ?

Question: When all the curves in a family \(G\left( {x,y,{c_1}} \right) = 0\)intersect orthogonally all the curves in another family\(H\left( {x,y,{c_2}} \right) = 0\), the families are said to be orthogonal trajectories of each other. See Figure 3.R.5. If \(dy/dx = f(x,y)\)is the differential equation of one family, then the differential equation for the orthogonal trajectories of this family is\(dy/dx = - 1/f(x,y)\). In Problems\(\;{\bf{15}} - {\bf{18}}\)find the differential equation of the given family by computing\(dy/dx\)and eliminating \({c_1}\)from this equation. Then find the orthogonal trajectories of the family. Use a graphing utility to graph both families on the same set of coordinate axes.

\({x^2} - 2{y^2} = {c_1}\)

Leaking Cylindrical Tank A tank in the form of a right circular cylinder standing on its end is leaking water through a circular hole in its bottom. As we saw in (10) of Section 1.3, when friction and contraction of water at the hole are ignored, the height h of water in the tank is described by

whereandare the cross-sectional areas of the water and the hole, respectively.

(a) Solve the DE if the initial height of the water isBy hand, sketch the graph ofand give its interval I of definition in terms of the symbolsand H. Use

(b) Suppose the tank is 10 feet high and has a radius of 2 feet and the circular hole has a radiusinch. If the tank is initially full, how long will it take to empty?
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