Transportation A plane flies from New York to Chicago (about...

Transportation A plane flies from New York to Chicago (about 700 miles) at a speed of 360 $\mathrm{mi} / \mathrm{h}$ . a. The speed $s$ of the plane is given by $s=\frac{d}{t}$ . $d$ represents the distance and $t$ is the time. Solve the equation for $t$ . b. Find the time for the trip. c. On the return trip from Chicago to New York, a tail wind helps the plane move faster. Write an expression for the speed of the plane on the return trip. Let $x$ represent the speed of the tail wind. d.The total flying time for the round trip is 3.5 h. Write a rational equatio for the sum of the flying times. Find the speed $x$ of the tail wind.

Transportation A plane flies from New York to Chicago (about 700 miles) at a speed of 360 $\mathrm{mi} / \mathrm{h}$ .
a. The speed $s$ of the plane is given by $s=\frac{d}{t}$ . $d$ represents the distance and $t$ is the time. Solve the equation for $t$ .
b. Find the time for the trip.
c. On the return trip from Chicago to New York, a tail wind helps the plane move faster. Write an expression for the speed of the plane on the return trip. Let $x$ represent the speed of the tail wind.
d.The total flying time for the round trip is 3.5 h. Write a rational equatio for the sum of the flying times. Find the speed $x$ of the tail wind.

Key Concepts

Distance-Time-Speed Relationship

This concept involves the fundamental formula connecting distance, speed, and time, expressed as speed = distance ÷ time. Understanding this relationship allows one to manipulate the formula to solve for any one of these variables when the other two are known, which is essential in motion and transportation problems.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging equations and isolating variables to solve for an unknown quantity. In this context, it is used to solve linear equations for time and form expressions for altered speeds when additional factors like a tail wind are included.

Rational Equations

Rational equations involve expressions that are fractions with polynomials in the numerator and denominator. They are crucial in modeling situations where time or rate is expressed as a ratio, such as when combining travel times for different parts of a journey and solving for unknown quantities like wind speed.

Tailwind Effect

The tailwind effect pertains to how an external factor—in this case, a wind blowing in the direction of travel—affects the overall speed of an object. It is modeled by adding the tail wind speed to the base speed, which is important in calculating adjusted travel times for journeys with varying conditions.

Transcript

00:01 This question starts off and says that a plane is flying from new york to chicago, which is about 700 miles, and it says that it's going a speed of 360 miles per hour.

00:13 It says the speed, or sorry, the speed can be written as d over t.

00:18 So it gives us this formula.

00:21 So s is d over t.

00:23 It says we're going 700 miles, and we're doing it at 360 miles per hour.

00:29 Okay, so that's the information that i just sort of said.

00:32 But part a says to solve this equation for t.

00:36 So for part a, all i have to say is that t is d over s.

00:41 Okay, these just switch places when you're solving that kind of scenario.

00:46 So that's my first answer.

00:47 Then it says for part b, find the time for the trip.

00:52 So the time is just going to be, he went 700 miles and 360 miles per hour.

01:01 So we just divide those and see what it gives us.

01:06 It's a little bit over 0 .5 right so it ends up being 35 over 18 and that is in hours so it's 35 over 18 hours all right now part c says that you get a tailwind on the way back and it says let x be the speed of the tailwind so it says right expression for the speed of the plane on the return trip so if you are going 360 miles an hour and you get a tail wind that helps you you you do just add x if x is the tailwind.

01:42 So that's how we do part c.

01:45 And then part d says, write an equation for the sum of the flying times and find the speed of the tailwind.

01:52 So what we know is the total time is going to be, sorry, 700 over 360 on the way there, and 700 over 360 plus x on the way back.

02:19 Okay.

02:23 Now it tells us, sorry i didn't say this, that the total flying time is three and a half hours...

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