Respuesta :
Answer:
[tex]0.83\text{ miles per minute}[/tex]
Step-by-step explanation:
GIVEN: In a [tex]10[/tex] mile race, Janet covered the first [tex]2[/tex] miles at a constant rate. She then speed up and rode her bike the last [tex]8[/tex] miles at a rate that was [tex]0.5[/tex] miles per minute faster. Janet's overall time would have been [tex]2[/tex] minutes faster had she ridden her bike the whole race at the faster pace.
TO FIND: What was Janet's average speed (in miles per minute) for the whole race?
SOLUTION:
let the speed of Janet in first [tex]2\text{ miles}[/tex] [tex]=x\text{ miles per minute}[/tex]
speed of Janet's bike in last [tex]8\text{ miles}[/tex] [tex]=x\text{+0.5 miles per minute}[/tex]
total time taken by Janet,
[tex]\text{Time}=\frac{\text{distance}}{\text{speed}}[/tex]
[tex]\text{Time taken}=\frac{2}{x}+\frac{8}{x+0.5}\text{minute}[/tex]
Time taken if Janet rides whole race at faster pace [tex]=\frac{10}{x+0.5}\text{minute}[/tex]
As, Janet's overall time would have been [tex]2\text{ minutes}[/tex] faster had she ridden her bike the whole race at the faster pace.
[tex]\frac{2}{x}+\frac{8}{x+0.5}=\frac{10}{x+0.5}+2[/tex]
[tex]\frac{2}{x}-\frac{2}{x+0.5}=2[/tex]
[tex]\frac{1}{x}-\frac{1}{x+0.5}=1[/tex]
[tex]x^{2} +0.5x-1=0[/tex]
on solving we get
[tex]x=0.5\text{ miles per minute}[/tex]
[tex]\text{Time taken}=\frac{2}{x}+\frac{8}{x+0.5}\text{minute}[/tex]
[tex]\text{Time taken}=\frac{2}{0.5}+\frac{8}{1}\text{minute}[/tex]
[tex]\text{Time taken}=\frac{2}{x}+\frac{8}{x+0.5}\text{minute}[/tex]
[tex]\text{Time taken}=12\text{minute}[/tex]
Average speed [tex]=\frac{\text{total distance}}{\text{time taken}}[/tex]
Average speed [tex]=\frac{10}{12}=0.83\text{miles per minute}[/tex]
Therefore average speed of Janet was [tex]0.83\text{ miles per minute}[/tex]
