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Applications of Rational Functions - Mathematics LibreTexts
depending on the direction of u remember at a certain time v will have to travel against u so hence myboat340 boatplans the boat would not b able to oppose the current. So speed of boat in still water finally means: it is the speed of the boat seen by an water molecule or an observer who is flowing with the water with the same speed as that of water current. It is analogous to the speed of a boy running inside a moving train as seen by a passenger inside the train. Now the speed of a boat while going upstream is actually the speed seen by an observer standing at rest on the bank of river. And speed of water current is again the speed seen by the same observer standing at rest on the bank of river.� see the water stream is running downside and putting the boat to back position thus boat speed would be reduced relative to myboat340 boatplans the speed of boat relative to water is bigger,therefore. V(bw) =V(bo) + V(wo). so, V(bo) =V(bw)-V(wo). The world unlimited water speed record is the officially recognised fastest speed achieved by a water-borne vehicle. The current record is km/h ( mph), achieved by Australian Ken Warby in the Spirit of Australia in The record is one of the sporting world's most hazardous competitions. Two official attempts to beat the record resulted in the death of the driver; despite this, there are several teams working to make further attempts.

In this section, we will investigate the use of rational functions in several applications. Note that the product of a number and its reciprocal is always equal to the number 1. Let x represent a nonzero number. To clear fractions from this equation, multiply both sides by the common denominator 10x. This equation is nonlinear it has a power of x larger than 1 , so make one side equal to zero by subtracting 29x from both sides of the equation.

Break up the middle term of the quadratic trinomial using this pair, then factor by grouping. Hence, we have two solutions for x. However, they both lead to the same number-reciprocal pair.

Note that. There are two numbers. The second number is 1 larger than twice the first number. Find the two numbers. Let x represent the first number.

Again, this equation is nonlinear. We will move everything to the right-hand side of this equation. Subtract 30x and 10 from both sides of the equation to obtain.

Break up the middle term using this pair and factor by grouping. Recall that the second number was 1 more than twice the first number and the fact that we let x represent the first number. That is, the second number is 5. If this is the first number, then the second number is. When we developed the Equations of Motion in the chapter on quadratic functions, we showed that if an object moves with constant speed, then the distance traveled is given by the formula.

A boat travels at a constant speed of 3 miles per hour in still water. In a river with unknown current, it takes the boat twice as long to travel 60 miles upstream against the current than it takes for the 60 mile return trip with the current.

What is the speed of the current in the river? The speed of the boat in still water is 3 miles per hour. When the boat travels upstream, the current is against the direction the boat is traveling and works to reduce the actual speed of the boat. When the boat travels downstream, then the actual speed of the boat is its speed in still water increased by the speed of the current.

Here is a useful piece of advice regarding distance, speed, and time tables. The third entry in each row is time. Therefore, the time of travel is. In similar fashion, the time to travel downstream is calculated with. To set up an equation, we need to use the fact that the time to travel upstream is twice the time to travel downstream. This leads to the result. This equation is linear no power of c other than 1.

Hence, we want to isolate all terms containing c on one side of the equation. We add c to both sides of the equation, then subtract from both sides of the equation. It is important to check that the solution satisfies the constraints of the problem statement. Note that the time to travel upstream 30 hours is twice the time to travel downstream 15 hours , so our solution is correct. A speedboat can travel 32 miles per hour in still water.

It travels miles upstream against the current then returns to the starting location. The total time of the trip is 10 hours. What is the speed of the current?

Let c represent the speed of the current. The trip each way is miles. Hence, the time it takes the boat to go upstream is given by. We can make the numbers a bit smaller by noting that both sides of the last equation are divisible by Discarding the negative answer speed is a positive quantity in this case , the speed of the current is 8 miles per hour.

Note that the total time to go upstream and return is 6. A nice application of rational functions involves the amount of work a person or team of persons can do in a certain amount of time. We can handle these applications involving work in a manner similar to the method we used to solve distance, speed, and time problems.

Here is the guiding principle. The amount of work done is equal to the product of the rate at which work is being done and the amount of time required to do the work.

That is,. For example, suppose that Emilia can mow lawns at a rate of 3 lawns per hour. After 6 hours,. A second important concept is the fact that rates add. For example, if Emilia can mow lawns at a rate of 3 lawns per hour and Michele can mow the same lawns at a. Bill can finish a report in 2 hours.

Maria can finish the same report in 4 hours. How long will it take them to finish the report if they work together? A common misconception is that the times add in this case. That is, it takes Bill 2 hours to complete the report and it takes Maria 4 hours to complete the same report, so if Bill and Maria work together it will take 6 hours to complete the report.

A little thought reveals that this result is nonsense. Clearly, if they work together, it will take them less time than it takes Bill to complete the report alone; that is, the combined time will surely be less than 2 hours. However, as we saw above, the rates at which they are working will add.

In our discussion above, we pointed out the fact that rates add. It takes Liya 7 more hours to paint a kitchen than it takes Hank to complete the same job. Together, they can complete the same job in 12 hours. How long does it take Hank to complete the job if he works alone?

Let H represent the time it take Hank to complete the job of painting the kitchen when he works alone. This last equation is nonlinear, so make one side zero by subtracting 24H and 84 from both sides of the equation. Find the number s. Moira can paddle her kayak at a speed of 2 mph in still water.

She paddles 3 miles upstream against the current and then returns to the starting location. The total time of the trip is 9 hours. What is the speed in mph of the current? Round your answer to the nearest hundredth. Boris is kayaking in a river with a 6 mph current.

Suppose that he can kayak 4 miles upstream in the same amount of time as it takes him to kayak 9 miles downstream. Jacob can paddle his kayak at a speed of 6 mph in still water. He paddles 5 miles upstream against the current and then returns to the starting location.

The total time of the trip is 5 hours. Boris can paddle his kayak at a speed of 6 mph in still water. If he can paddle 5 miles upstream in the same amount of time as it takes his to paddle 9 miles downstream, what is the speed of the current? Jacob is canoeing in a river with a 5 mph current.

Suppose that he can canoe 4 miles upstream in the same amount of time as it takes him to canoe 8 miles downstream. The speed of a freight train is 16 mph slower than the speed of a passenger train. The passenger train travels miles in the same time that the freight train travels miles. Find the speed of the freight train. The speed of a freight train is 20 mph slower than the speed of a passenger train. Emily can paddle her canoe at a speed of 2 mph in still water.

She paddles 5 miles upstream against the current and then returns to the starting location. The total time of the trip is 6 hours. Jacob is canoeing in a river with a 2 mph current. Suppose that he can ca- noe 2 miles upstream in the same amount of time as it takes him to canoe 5 miles downstream. If she can paddle 4 miles upstream in the same amount of time as it takes her to paddle 8 miles downstream, what is the speed of the current?

If he can paddle 5 miles upstream in the same amount of time as it takes his to paddle 10 miles downstream, what is the speed of the current? The speed of a freight train is 19 mph slower than the speed of a passenger train. It takes Jean 15 hours longer to complete an inventory report than it takes Sanjay.

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