Amanda, Bryce, and Corey enter a race in which they have to...

Amanda, Bryce, and Corey enter a race in which they have to run, swim, and cycle over a marked course. Their average speeds are given in the table. Corey finishes first with a total time of $1 \mathrm{h} 45 \mathrm{min}$. Amanda comes in second with a time of $2 \mathrm{h} 30 \mathrm{min}$. Bryce finishes last with a time of 3 h. Find the distance (in $m$ i) for each part of the race.

Key Concepts

Distance-Speed-Time Relationship

This fundamental concept relates distance, speed, and time through the equation Distance = Speed × Time. It is critical for solving problems where different segments of an event are governed by specific speeds and durations, allowing the determination of one variable if the others are known.

Unit Conversion

Accurate calculations often require consistent units, so converting minutes to hours (or vice versa) is essential when speeds are given in one time unit and times in another. This ensures that all terms in the equations are compatible, avoiding calculation errors.

Systems of Linear Equations

In problems where multiple unknowns need to be determined simultaneously from different sets of data, setting up and solving a system of linear equations is key. The overall event is divided into segments, and relationships between distance, speed, and time for each segment are expressed as equations that together form a system.

Multi-Discipline Race Analysis

This concept involves breaking down a race into distinct sections—such as running, swimming, and cycling—each with its own distance and speed. The overall race time is the sum of the times for each segment, and understanding this breakdown is crucial for modeling the problem and solving for the unknown distances.

Transcript

00:01 Let running distance let running distance x swimming distance y and cycling cycling distance equal to jet now it is saying that it is given the speed get the speed into speed what? it is given this speed to distance time so how much time will take for running? 10 into x how much time? for amanda swimming 4 into y for amanda cycling 20 into z so that is total time amanda how much time takes so the amanda takes amanda comes 2 hour and 30 minutes so actually i'm writing in the minute that is the best right so write answer as okay it is given hours so two it is given hours so you write in the hours so you write in the only so 2 hours 30 minute which is 5 by 2 hours right so this is equation 1 now this is 15 by 2 x plus 6 y plus 15 chat which is equal to brisk how much time takes so brief finishes 3 hours in 3 hours and what about this correy, 15x plus 3y plus 40 z equal to its corrective 1 or 45 minutes.

02:00 Means 1 and 3 by 4.

02:04 So basically 7 by 4.

02:08 So this equation 3.

02:13 And now solve this and find the value of x, y, z and that will be the answer.

02:17 So let us solve this linear equation.

02:20 So multiply with 2 in the equation 2.

02:26 So multiply with 2 in equation 2.

02:32 So we find 15x plus 12 y, sorry, 12 y.

02:41 12 y plus become 30.

02:45 So just remove this.

02:50 This is 15x plus 12 o.

02:55 Y plus 30 z equal to 6 and third it is given 15 x plus 3 y plus 40 z equal to 7 by 4 now subtract this sandwich this is cancelled out 12 y minus 3 how much 9y and this is minus 10 z equal to this is 17 by 4 so say this equation over 4 now what you do that cancel x from 1 and 3 so what is the best thing you can do that multiply with here you might here multiply with 3 and multiply with 2 in the equation number 3 so we'll find 30x plus 12 y 30 x plus 12 y multiply me 3 and plus 60 z this is 15 by 2 and multiply the third by 2 so this is 30x plus y how much it is 6y 6y z how much it is 40 z then it is 80 z then it is 80 x equal to 7 by 2 now subtract this...

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