/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 73 An executive flew in the corpora... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 73

An executive flew in the corporate jet to a meeting in a city 1500 kilometers away. After traveling the same amount of time on the return flight, the pilot mentioned that they still had 300 kilometers to go. The air speed of the plane was 600 kilometers per hour. How fast was the wind blowing? (Assume that the wind direction was parallel to the flight path and constant all day.)

Short Answer

Expert verified
The speed of the wind was approximately 54.54 kilometers per hour.

Step by step solution

01

Understanding the Problem

The actual speed of the jet is compounded of two speeds: its own speed and the wind's speed. When moving against the wind, these speeds are subtracted from each other, resulting in a slower speed overall. Conversely, when moving with the wind, these speeds are added together, resulting in a faster overall speed. The time taken by the plane to cover certain distances at differing speeds due to changes in wind direction allows us to determine the speed of the wind.
02

Setting up the equations

As stated in the prompt, the plane flew for the same amount of time each way. Therefore, the time it took to travel the 1500 kilometers with the wind plus the time it took to travel the 1200 kilometers (1500 kilometers there, 300 kilometers more to go) against the wind equals to twice the time it took to travel with the wind alone. If we use \(w\) to represent the speed of the wind, this can be set up as the following equation: \[\frac{1800}{600-w} = \frac{1500}{600+w}\]
03

Solving for the wind speed

Cross multiply and solve for \(w\). This gives: \[1800(600+w) = 1500(600-w)\]. Expand the brackets and solve for \(w\). This gives: \[1080000 + 1800w = 900000 - 1500w\]. Combine the \(w\) terms, this gives: \[3300w = 180000\]. Finally, divide by 3300 to isolate \(w\), which gives: \[w = \frac{180000}{3300} \approx 54.54\] kilometers per hour. So the speed of the wind was approximately 54.54 kilometers per hour.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Equations
In mathematics, equations are essential tools. They represent relationships between different quantities or variables. By setting two expressions equal to each other, equations help us find unknown values. In the jet flight problem, we used an equation to find the wind speed. Understanding how to construct equations from word problems is a critical algebra skill.

To set up an equation, we first identify what we want to find and assign a variable to it, often labeled as a letter like \(w\) for wind speed. Then, based on the conditions provided in the problem, we set equal the expressions involving this unknown. In our exercise, the time taken for the plane in both directions was key. This allowed us to create a single equation accounting for both the outbound and inbound journey. Making sure the units are consistent and interpreting the problem's language into mathematical symbols is crucial in accurately setting up equations.
Distance-Speed-Time Relationships
Understanding the foundation of distance-speed-time relationships is crucial in solving motion problems like the jet flight exercise. The basic formula to remember is:
  • Distance = Speed \(\times\) Time
  • Speed = \(\frac{\text{Distance}}{\text{Time}}\)
  • Time = \(\frac{\text{Distance}}{\text{Speed}}\)
In our problem, the same time constraint played a vital role, meaning the jet's outbound trip (with wind) and the inbound trip (against wind) each took the same amount of time. The variable wind speed affects how quickly the plane travels.

The exercise uses the distances (1500 km for outbound, 1200 km for inbound after the return) and the plane's air speed (600 km/h) to create time-related equations. By substituting these into our distance-speed-time formulas, we bridge the gap between the physical scenario and mathematical technique. Recognizing these relationships enables us to interpret complex motion problems efficiently.
Cross Multiplication
Cross multiplication is an algebraic technique used to solve equations involving fractions. When we have an equation of the form \( \frac{A}{B} = \frac{C}{D} \), we can find unknowns by cross multiplying 鈥� multiplying the numerator of each fraction by the denominator of the other. This simplifies the equation, often eliminating the fractions, which makes it easier to solve.

In the jet problem, cross multiplication was used on the equation \( \frac{1800}{600-w} = \frac{1500}{600+w} \). By cross multiplying, we obtained the equation \(1800(600+w) = 1500(600-w)\), free from fractions, and solvable by basic algebraic manipulation. This step is crucial for handling any fractions resulting from division in distance-speed-time problems and simplifies solving for our desired variable. Once cross multiplication is complete, solving the now linear equation becomes straightforward by using basic algebra to combine like terms and isolate the variable, giving us the answer we're looking for.

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Most popular questions from this chapter

Solve the inequality and graph the solution on the real number line. Use a graphing utility to verify your solution graphically. $$2 x^{3}+3 x^{2}<11 x+6$$

Find the domain of \(x\) in the expression. $$\sqrt{-x^{2}+x+12}$$

Solving an Equation Involving an Absolute Value Find all solutions of the equation algebraically. Check your solutions. $$|x+1|=x^{2}-5$$

You want to determine whether there is a relationship between an athlete's weight \(x\) (in pounds) and the athlete's maximum bench-press weight \(y\) (in pounds). Sample data from 12 athletes are shown below. (Spreadsheet at LarsonPrecalculus.com) (165,170),(184,185),(150,200), (210,255),(196,205),(240,295), (202,190),(170,175),(185,195), (190,185),(230,250),(160,150) (a) Use a graphing utility to plot the data. (b) A model for the data is \(y=1.3 x-36\).Use the graphing utility to graph the equation in the same viewing window used in part (a). (c) Use the graph to estimate the values of \(x\) that predict a maximum bench- press weight of at least 200 pounds. (d) Use the graph to write a statement about the accuracy of the model. If you think the graph indicates that an athlete's weight is not a good indicator of the athlete's maximum bench-press weight, list other factors that might influence an individual's maximum bench-press weight.

Graphical Analysis (a) use a graphing utility to graph the equation, (b) use the graph to approximate any \(x\) -intercepts of the graph, (c) set \(y=0\) and solve the resulting equation, and (d) compare the result of part (c) with the \(x\) -intercepts of the graph. $$y=x+\frac{9}{x+1}-5$$
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