Lesson Wind and Current problems
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<H2>Wind and Current problems</H2> In this lesson some typical <B>Travel and Distance</B> problems of the type <B>Wind and Current</B> are presented for a motorboat and airplane making round trips. In <B>problem 1</B> and <B>2</B> the length of the trip is given, as well as spent time moving in each direction. A motorboat (airplane) speed in still water (still air) and the current (wind) speed are unknown. The way to solve these problems is to reduce them to the system of two linear equations with two unknowns, and then to solve this system. In <B>problem 3</B> and <B>4</B> the duration of the trip is given to each direction, as well as the one of two speeds. The other speed and the travel length are unknown. The way to solve these problems is to reduce them to one linear equation with one unknown variable - speed, and then to solve this equation. When it is done, you can calculate the length of the trip. <H3>Problem 1. Motorboat moving upstream and downstream on a river</H3>A motorboat makes the 24 miles upstream trip on a river against the current in 3 hours. Returning trip with the current takes 2 hours. Find the motorboat speed in still water and the current speed. <B> Solution</B> Let u be the motorboat speed in still water in miles per hour, and v be the current speed in miles per hour. Then the speed of the motorboat is u - v relative to the river's banks when it moves upstream, and u + v when it is moves downstream. For the upstream trip you have the equation connecting the speed, the time and the distance in the form 3*(u - v) = 24. For the downstream trip you have similar equation in the form 2*(u + v) = 24. Thus you have the system of two linear equations in two unknowns {{{system(3(u - v) = 24, 2(u + v) = 24 )}}}. In the first equation divide both sides by 3. In the second equation divide both sides by 2. You will get an equivalent system {{{system(u - v = 8, u + v = 12 )}}}. Add the first and the second equations. You will get 2u = 8 + 12 = 20. Hence, u = {{{20/2}}} = 10 miles per hour. Now, substitute this value of u into equation u + v = 12. You will get v = 12 - 10 = 2 miles per hour. <B>Answer</B>. The motorboat speed in still water is equal to 10 miles per hour. The current speed is 2 miles per hour. <H3>Problem 2. Airplane flying into the wind and with the wind</H3>When an airplane flies into the wind, it can travel 3000 miles in 6 hours. When it flies with the wind, it can travel the same distance in 5 hours. Find the speed of the airplane in still air and the speed of the wind. <B> Solution</B> Let u be the airplane speed in still air in miles per hour, and v be the speed of the wind in miles per hour. Then the speed of the airplane is u - v relative the earth when it moves into the wind, and u + v when it moves with the wind. For the flight into the wind you have the equation connecting the speed, the time and the distance in the form 6*(u - v) = 3000. For the flight with the wind you have similar equation in the form 5*(u + v) = 3000. Thus you have the system of two linear equations with two unknowns {{{system(6(u - v) =3000, 5(u + v) = 3000 )}}}. In the first equation divide both sides by 6. In the second equation divide both sides by 5. You will get an equivalent system {{{system(u - v = 500, u + v = 600 )}}}. Add the first and the second equations. You will get 2u = 500 + 600 = 1100. Hence, u = {{{1100/2}}} = 550 miles per hour. Now, substitute this value of u into equation u + v = 600. You will get v = 600 - 550 = 50 miles per hour. <B>Answer</B>. The speed of the airplane in still air is equal to 550 miles per hour. The speed of the wind is 50 miles per hour. <H3>Problem 3. Motorboat moving upstream and downstream on a river</H3>A motorboat makes an upstream trip on a river in 3 hours against the current, which is of 2 miles per hour. The return downstream trip with the same current takes 2 hours. Find the motorboat speed in still water and the trip length. <B> Solution</B> Let u denote the motorboat speed in still water in miles per hour. Then the speed of the motorboat is u - 2 relative to the river's banks when it moves upstream, and u + 2 when it moves downstream. The length of the upstream trip is equal to 3*(u - 2) miles. The length of the downstream trip is equal to 2*(u + 2) miles. Since it is the same length, this gives you an equation with one unknown 3*(u - 2) = 2*(u + 2). Let's open parentheses, collect variable terms on the left side, constant terms on the right side and reduce like terms, step by step: 3u - 6 = 2u + 4, 3u - 2u = 6 + 4, u = 10. Thus we found the motorboat speed in still water as 10 miles per hour. Now, determine the trip length by substituting u=10 into the formula L = 3*(10 - 2) = 3*8 = 24 miles. <B>Answer</B>. The motorboat speed in still water is equal to 10 miles per hour. The trip length is equal to 24 miles. <H3>Problem 4. Airplane flying into the wind and with the wind</H3> Airplane flies for 6 hours against the wind. The return fly with the same tail wind takes 5 hours. The airplane speed in the still air is 550 miles per hour. Find the wind speed and the fly length. <B> Solution</B> Let u be the wind speed in miles per hour. Then the speed of the airplane is 550 - v when it moves against the wind, and 550 + v when it moves with the wind. The airplane travels for 6*(550-v) miles when flies against the wind. The airplane travels for 5*(550+v) miles when flies with the wind. Since it is the same length, this gives us the equation with one unknown 6*(550 - v) = 5*(550 + v). Let's open parentheses, collect variable terms on the left side, constant terms on the right side and reduce like terms, step by step: 3300 - 6v = 2750 + 5v, 3300 - 2750 = 5v + 6v, 11v = 550, v = 50. Thus we found the wind speed as 50 miles per hour. Now, determine the trip length by substituting v=50 into the formula L = 6(550 - v) = 6*500 = 3000 miles. <B>Answer</B>. The wind speed is equal to 50 miles per hour. The fly length is equal to 3000 miles. My other lessons on <B>Travel and Distance</B> problems in this site are <TABLE> <TR> <TD> - <A HREF=http://www.algebra.com/algebra/homework/word/travel/Travel-and-Distance-problems.lesson>Travel and Distance problems</A> - <A HREF=https://www.algebra.com/algebra/homework/word/travel/Travel-and-Distance-problems-for-two-bodies-moving-toward-each-other.lesson>Travel and Distance problems for two bodies moving in opposite directions</A> - <A HREF=https://www.algebra.com/algebra/homework/word/travel/Typical-catching-up-Travel-and-Distance-problems.lesson>Travel and Distance problems for two bodies moving in the same direction (catching up)</A> - <A HREF=http://www.algebra.com/algebra/homework/NumericFractions/Using-fractions-to-solve-Travel-problems.lesson>Using fractions to solve Travel problems</A> - <A HREF=http://www.algebra.com/algebra/homework/word/travel/More-problems-on-upstream-and-downstream-round-trips.lesson>More problems on upstream and downstream round trips</A> - <A HREF=http://www.algebra.com/algebra/homework/word/travel/Wind-and-Current-problems-solvable-by-quadratic-equations.lesson>Wind and Current problems solvable by quadratic equations</A> - <A HREF=https://www.algebra.com/algebra/homework/word/travel/Unpowered-raft-moving-downstream-along-a-river.lesson>Unpowered raft floating downstream along a river</A> - <A HREF=https://www.algebra.com/algebra/homework/word/travel/Selected-problems-from-the-archive-on-a-boat-floating-Upstream-and-Downstream.lesson>Selected problems from the archive on the boat floating Upstream and Downstream</A> - <A HREF=https://www.algebra.com/algebra/homework/word/travel/Selected-problems-from-the-archive-on-a-plane-flying-with-and-against-the-wind.lesson>Selected problems from the archive on a plane flying with and against the wind</A> - <A HREF=https://www.algebra.com/algebra/homework/word/travel/Selected-Travel-and-Distance-problems-from-the-archive.lesson>Selected Travel and Distance problems from the archive</A> - <A HREF=https://www.algebra.com/algebra/homework/word/travel/Had-a-car-move-faster-it-would-arrive-quicker.lesson>Had a car move faster it would arrive sooner</A> - <A HREF=http://www.algebra.com/algebra/homework/word/travel/How-far-do-you-live-from-school.lesson>How far do you live from school?</A> - <A HREF=http://www.algebra.com/algebra/homework/word/travel/One-unusual-Travel-problem.lesson>One unusual Travel problem</A> - <A HREF=http://www.algebra.com/algebra/homework/word/travel/Another-unusual-Travel-problem.lesson>Another unusual Travel problem (Arnold's problem on two walking old women)</A> - <A HREF=http://www.algebra.com/algebra/homework/word/travel/Travel-problem-on-a-messenger-moving-back-and-forth-along-the-marching-army-column.lesson>Travel problem on a messenger moving back and forth along the marching army's column</A> - <A HREF=https://www.algebra.com/algebra/homework/word/travel/A-person-walking-along-the-street-and-the-buses-traveling-in-the-same-and-opposite-directions.lesson>A person walking along the street and buses traveling in the same and opposite directions</A> </TD> <TD> - <A HREF=http://www.algebra.com/algebra/homework/word/travel/Calculating-an-average-speed.lesson>Calculating an average speed: a train going from A to B and back</A> - <A HREF=http://www.algebra.com/algebra/homework/word/travel/One-more-problem-on-calculating-an-average-speed.lesson>One more problem on calculating an average speed</A> - <A HREF=http://www.algebra.com/algebra/homework/word/travel/Clock-problems.lesson>Clock problems</A> - <A HREF=https://www.algebra.com/algebra/homework/word/travel/Advanced-clock-problems.lesson>Advanced clock problems</A> - <A HREF=http://www.algebra.com/algebra/homework/word/travel/Problems-on-bodies-moving-on-a-circle.lesson>Problems on bodies moving on a circle</A> - <A HREF=http://www.algebra.com/algebra/homework/word/travel/A-train-passing-a-telegraph-post-and-passing-a-bridge.lesson>A train passing a telegraph post and passing a bridge</A> - <A HREF=http://www.algebra.com/algebra/homework/word/travel/A-train-passing-a-platform.lesson>A train passing a platform</A> - <A HREF=http://www.algebra.com/algebra/homework/word/travel/A-train-passing-a-tunnel.lesson>A train passing through a tunnel</A> - <A HREF=http://www.algebra.com/algebra/homework/word/travel/A-light-rail-train-passing-a-walking-person.lesson>A light-rail train passing a walking person</A> - <A HREF=http://www.algebra.com/algebra/homework/word/travel/A-train-passing-another-train.lesson>A train passing another train</A> - <A HREF=https://www.algebra.com/algebra/homework/word/travel/A-man-crossing-a-bridge-when-a-train-comes-from-behind.lesson>A man crossing a bridge and a train coming from behind</A> - <A HREF=https://www.algebra.com/algebra/homework/word/travel/27-A-problem-on-a-rower-going-on-a-river-who-missed-the-bottle-of-whiskey-under-a-bridge.lesson>A rower going on a river who missed the bottle of whiskey under a bridge</A> - <A HREF=https://www.algebra.com/algebra/homework/word/travel/Non-traditional-Travel-and-Distance-problems.lesson>Non-traditional Travel and Distance problems</A> - <A HREF=https://www.algebra.com/algebra/homework/word/travel/The-distance-covered-by-a-free-falling-body-during-last-second-of-the-fall.lesson>The distance covered by a free falling body during last second of the fall</A> - <A HREF=https://www.algebra.com/algebra/homework/word/travel/The-Doppler-shift.lesson>The Doppler Shift</A> - <A HREF=https://www.algebra.com/algebra/homework/word/travel/Entertainment-Travel-and-Distance-problems.lesson>Entertainment Travel and Distance problems</A> - <A HREF=https://www.algebra.com/algebra/homework/word/travel/OVERVIEW-of-lessons-on-Travel-and-Distance.lesson>OVERVIEW of lessons on Travel and Distance</A> </TD> </TR> </TABLE> Use this file/link <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A> to navigate over all topics and lessons of the online textbook ALGEBRA-I.
