8th Edition

ISBN: 9781285740621

Author: James Stewart

Publisher: Cengage Learning

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Textbook Question

Chapter 13.4, Problem 33E

Water traveling along a straight portion of a river normally flows fastest in the middle, and the speed slows to almost zero at the banks. Consider a long straight stretch of river flowing north, with parallel banks 40 m apart. If the maximum water speed is 3 m/s, we can use a quadratic function as a basic model for the rate of water flow x units from the west bank: f ( x ) = 3 400 x ( 40 − x ) .

(a) A boat proceeds at a constant speed of 5 m/s from a point A on the west bank while maintaining a heading perpendicular to the bank. How far down the river on the opposite bank will the boat touch shore? Graph the path of the boat.

(b) Suppose we would like to pilot the boat to land at the point B on the east bank directly opposite A. If we maintain a constant speed of 5 m/s and a constant heading, find the angle at which the boat should head. Then graph the actual path the boat follows. Does the path seem realistic?

Expert Solution & Answer

Chapter 13 Solutions

Calculus (MindTap Course List)

Ch. 13.1 - Prob. 1E Ch. 13.1 - Prob. 2E Ch. 13.1 - Prob. 3E Ch. 13.1 - Find the limit. limt1(t2tt1i+t+8j+sintlntk)Ch. 13.1 - Prob. 5E Ch. 13.1 - Prob. 6E Ch. 13.1 - Sketch the curve with the given vector equation....Ch. 13.1 - Prob. 8E Ch. 13.1 - Prob. 9E Ch. 13.1 - Prob. 10E

Ch. 13.1 - Prob. 11E Ch. 13.1 - Prob. 12E Ch. 13.1 - Prob. 13E Ch. 13.1 - Prob. 14E Ch. 13.1 - Prob. 15E Ch. 13.1 - Prob. 16E Ch. 13.1 - Prob. 17E Ch. 13.1 - Prob. 18E Ch. 13.1 - Prob. 19E Ch. 13.1 - Prob. 20E Ch. 13.1 - Prob. 21E Ch. 13.1 - Match the parametric equations with the graphs...Ch. 13.1 - Match the parametric equations with the graphs...Ch. 13.1 - Match the parametric equations with the graphs...Ch. 13.1 - Prob. 25E Ch. 13.1 - Prob. 26E Ch. 13.1 - Prob. 27E Ch. 13.1 - Prob. 28E Ch. 13.1 - Prob. 29E Ch. 13.1 - Prob. 30E Ch. 13.1 - Prob. 31E Ch. 13.1 - Prob. 32E Ch. 13.1 - Prob. 33E Ch. 13.1 - Prob. 34E Ch. 13.1 - Prob. 35E Ch. 13.1 - Use a computer to graph the curve with the given...Ch. 13.1 - Use a computer to graph the curve with the given...Ch. 13.1 - Graph the curve with parametric equations...Ch. 13.1 - Graph the curve with parametric equations...Ch. 13.1 - Prob. 40E Ch. 13.1 - Show that the curve with parametric equations...Ch. 13.1 - Prob. 42E Ch. 13.1 - Prob. 43E Ch. 13.1 - Prob. 44E Ch. 13.1 - Prob. 45E Ch. 13.1 - Prob. 46E Ch. 13.1 - Try to sketch by hand the curve of intersection of...Ch. 13.1 - Try to sketch by hand the curve of intersection of...Ch. 13.1 - If two objects travel through space along two...Ch. 13.1 - Prob. 50E Ch. 13.1 - a Graph the curve with parametric equations...Ch. 13.1 - Prob. 52E Ch. 13.1 - Prob. 53E Ch. 13.1 - Prob. 54E Ch. 13.2 - Prob. 1E Ch. 13.2 - Prob. 2E Ch. 13.2 - Prob. 3E Ch. 13.2 - Prob. 4E Ch. 13.2 - Prob. 5E Ch. 13.2 - Prob. 6E Ch. 13.2 - Prob. 7E Ch. 13.2 - Prob. 8E Ch. 13.2 - Prob. 9E Ch. 13.2 - Prob. 10E Ch. 13.2 - Prob. 11E Ch. 13.2 - Prob. 12E Ch. 13.2 - Prob. 13E Ch. 13.2 - Prob. 14E Ch. 13.2 - Prob. 15E Ch. 13.2 - Prob. 16E Ch. 13.2 - Prob. 17E Ch. 13.2 - Prob. 18E Ch. 13.2 - Prob. 19E Ch. 13.2 - Prob. 20E Ch. 13.2 - Prob. 21E Ch. 13.2 - Prob. 22E Ch. 13.2 - Prob. 23E Ch. 13.2 - Prob. 24E Ch. 13.2 - Prob. 25E Ch. 13.2 - Prob. 26E Ch. 13.2 - Prob. 27E Ch. 13.2 - Prob. 28E Ch. 13.2 - Prob. 29E Ch. 13.2 - Find parametric equations for the tangent line to...Ch. 13.2 - Prob. 31E Ch. 13.2 - Prob. 32E Ch. 13.2 - Prob. 33E Ch. 13.2 - Prob. 34E Ch. 13.2 - Evaluate the integral. 02(tit3j+3t5k)dt Ch. 13.2 - Prob. 36E Ch. 13.2 - Prob. 37E Ch. 13.2 - Prob. 38E Ch. 13.2 - Evaluate the integral. (sec2ti+t(t2+1)3j+t2lntk)dt Ch. 13.2 - Prob. 40E Ch. 13.2 - Prob. 41E Ch. 13.2 - Prob. 42E Ch. 13.2 - Prob. 43E Ch. 13.2 - Prove Formula 3 of Theorem 3.Ch. 13.2 - Prove Formula 5 of Theorem 3.Ch. 13.2 - Prob. 46E Ch. 13.2 - Prob. 47E Ch. 13.2 - If u and v are the vector functions in Exercise...Ch. 13.2 - Prob. 49E Ch. 13.2 - Prob. 50E Ch. 13.2 - If r(t)=acost+bsint, where a and b are constant...Ch. 13.2 - Prob. 52E Ch. 13.2 - Prob. 53E Ch. 13.2 - Find an expression for ddt[u(t)(v(t)w(t))].Ch. 13.2 - Prob. 55E Ch. 13.2 - Prob. 56E Ch. 13.2 - Prob. 57E Ch. 13.2 - Prob. 58E Ch. 13.3 - Find the length of the curve....Ch. 13.3 - Prob. 2E Ch. 13.3 - Prob. 3E Ch. 13.3 - Prob. 4E Ch. 13.3 - Find the length of the curve. r(t)=i+t2j+t3k,0t1 Ch. 13.3 - Prob. 6E Ch. 13.3 - Prob. 7E Ch. 13.3 - Find the length of the curve correct of four...Ch. 13.3 - Prob. 9E Ch. 13.3 - Graph the curve with parametric equations...Ch. 13.3 - Let C be the curve of intersection of the...Ch. 13.3 - Find, correct to four decimal places, the length...Ch. 13.3 - a Find the arc length function for the curve...Ch. 13.3 - a Find the arc length function for the curve...Ch. 13.3 - Prob. 15E Ch. 13.3 - Reparametrize the curve r(t)=(2t2+11)i+2tt2+1j...Ch. 13.3 - a Find the unit tangent and unit normal vectors...Ch. 13.3 - Prob. 18E Ch. 13.3 - Prob. 19E Ch. 13.3 - Prob. 20E Ch. 13.3 - Use Theorem 10 to find the curvature. r(t)=t3j+t2k Ch. 13.3 - Use Theorem 10 to find the curvature....Ch. 13.3 - Prob. 23E Ch. 13.3 - Find the curvature of r(t)=t2,lnt,tlnt at the...Ch. 13.3 - Find the curvature of r(t)=t,t2,t3 at the point...Ch. 13.3 - Graph the curve with parametric equations...Ch. 13.3 - Use Formula 11 to find the curvature. y=x4 Ch. 13.3 - Prob. 28E Ch. 13.3 - Use Formula 11 to find the curvature. y=xex Ch. 13.3 - Prob. 30E Ch. 13.3 - Prob. 31E Ch. 13.3 - Find an equation of a parabola that has curvature...Ch. 13.3 - a Is the curvature of the curve C shown in the...Ch. 13.3 - Prob. 34E Ch. 13.3 - Prob. 35E Ch. 13.3 - Prob. 36E Ch. 13.3 - Prob. 37E Ch. 13.3 - Two graphs, a and b, are shown. One is a curve...Ch. 13.3 - Two graphs, a and b, are shown. One is a curve...Ch. 13.3 - Prob. 40E Ch. 13.3 - Prob. 41E Ch. 13.3 - Prob. 42E Ch. 13.3 - Prob. 43E Ch. 13.3 - Prob. 44E Ch. 13.3 - Prob. 45E Ch. 13.3 - Prob. 46E Ch. 13.3 - Prob. 47E Ch. 13.3 - Prob. 48E Ch. 13.3 - Find equations of the normal plane and osculating...Ch. 13.3 - Find equations of the normal plane and osculating...Ch. 13.3 - Find equations of the osculating circles of the...Ch. 13.3 - Find equations of the osculating circles of the...Ch. 13.3 - Prob. 53E Ch. 13.3 - Is there a point on the curve in Exercise 53 where...Ch. 13.3 - Find equations of the normal and osculating planes...Ch. 13.3 - Prob. 56E Ch. 13.3 - Show that at every point on the curve...Ch. 13.3 - Prob. 58E Ch. 13.3 - Prob. 59E Ch. 13.3 - Prob. 60E Ch. 13.3 - a Show that dB/ds is perpendicular to B. b Show...Ch. 13.3 - Prob. 62E Ch. 13.3 - Use the Frenet-Serret formulas to prove each of...Ch. 13.3 - Show that the circular helix r(t)=acost,asint,bt,...Ch. 13.3 - Prob. 65E Ch. 13.3 - Prob. 66E Ch. 13.3 - Prob. 67E Ch. 13.3 - Prob. 68E Ch. 13.4 - The table gives coordinates of a particle moving...Ch. 13.4 - The figure shows the path of a particle that moves...Ch. 13.4 - Prob. 3E Ch. 13.4 - Prob. 4E Ch. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Prob. 6E Ch. 13.4 - Prob. 7E Ch. 13.4 - Prob. 8E Ch. 13.4 - Prob. 9E Ch. 13.4 - Prob. 10E Ch. 13.4 - Prob. 11E Ch. 13.4 - Prob. 12E Ch. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Find the velocity, acceleration, and speed of a...Ch. 13.4 - Prob. 15E Ch. 13.4 - Prob. 16E Ch. 13.4 - a Find the position vector of a particle that has...Ch. 13.4 - Prob. 18E Ch. 13.4 - The position function of a particle is given by...Ch. 13.4 - Prob. 20E Ch. 13.4 - A force with magnitude 20 N acts directly upward...Ch. 13.4 - Show that if a particle moves with constant speed,...Ch. 13.4 - A projectile is fired with an initial speed of 200...Ch. 13.4 - Prob. 24E Ch. 13.4 - Prob. 25E Ch. 13.4 - A projectile is fired from a tank with initial...Ch. 13.4 - A rifle is fired with angle of elevation 36. What...Ch. 13.4 - A batter hits a baseball 3 ft above the ground...Ch. 13.4 - A medieval city has the shape of a square and is...Ch. 13.4 - Show that a projectile reaches three-quarters of...Ch. 13.4 - A ball is thrown eastward into the air from the...Ch. 13.4 - Prob. 32E Ch. 13.4 - Water traveling along a straight portion of a...Ch. 13.4 - Prob. 34E Ch. 13.4 - Prob. 35E Ch. 13.4 - Prob. 36E Ch. 13.4 - Prob. 37E Ch. 13.4 - Prob. 38E Ch. 13.4 - Prob. 39E Ch. 13.4 - Prob. 40E Ch. 13.4 - Find the tangential and normal components of the...Ch. 13.4 - Prob. 42E Ch. 13.4 - The magnitude of the acceleration vector a is 10...Ch. 13.4 - Prob. 44E Ch. 13.4 - The position function of a spaceship is...Ch. 13.4 - Prob. 46E Ch. 13.R - Prob. 1CC Ch. 13.R - Prob. 2CC Ch. 13.R - Prob. 3CC Ch. 13.R - Prob. 4CC Ch. 13.R - Prob. 5CC Ch. 13.R - Prob. 6CC Ch. 13.R - Prob. 7CC Ch. 13.R - Prob. 8CC Ch. 13.R - Prob. 9CC Ch. 13.R - Prob. 1TFQ Ch. 13.R - Prob. 2TFQ Ch. 13.R - Prob. 3TFQ Ch. 13.R - Prob. 4TFQ Ch. 13.R - Prob. 5TFQ Ch. 13.R - Prob. 6TFQ Ch. 13.R - Determine whether the statement is true or false....Ch. 13.R - Prob. 8TFQ Ch. 13.R - Prob. 9TFQ Ch. 13.R - Prob. 10TFQ Ch. 13.R - Prob. 11TFQ Ch. 13.R - Prob. 12TFQ Ch. 13.R - Prob. 13TFQ Ch. 13.R - Prob. 14TFQ Ch. 13.R - Prob. 1E Ch. 13.R - Prob. 2E Ch. 13.R - Prob. 3E Ch. 13.R - Prob. 4E Ch. 13.R - Prob. 5E Ch. 13.R - Prob. 6E Ch. 13.R - Prob. 7E Ch. 13.R - Prob. 8E Ch. 13.R - Prob. 9E Ch. 13.R - Prob. 10E Ch. 13.R - For the curve given by r(t)=sin3t,cos3t,sin2t,...Ch. 13.R - Find the curvature of the ellipse x=3cost,y=4sint...Ch. 13.R - Find the curvature of the curve y=x4 at the point...Ch. 13.R - Find an equation of the osculating circle of the...Ch. 13.R - Prob. 15E Ch. 13.R - The figure shows the curve C traced by a particle...Ch. 13.R - A particle moves with position function...Ch. 13.R - Prob. 18E Ch. 13.R - A particle starts at the origin with initial...Ch. 13.R - Prob. 20E Ch. 13.R - A projectile is launched with an initial speed of...Ch. 13.R - Prob. 22E Ch. 13.R - Prob. 23E Ch. 13.R - In designing transfer curves to connect sections...Ch. 13.P - A particle P moves with constant angular speed ...Ch. 13.P - A circular curve of radius R on a highway is...Ch. 13.P - A projectile is fired from the origin with angle...Ch. 13.P - a A projectile is fired from the origin down an...Ch. 13.P - A ball rolls off a table with a speed of 2 ft/s....Ch. 13.P - Prob. 6P Ch. 13.P - If a projectile is fired with angle of elevation ...Ch. 13.P - Prob. 8P Ch. 13.P - Prob. 9P

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