Respuesta :
Answer:
They will meet at a distance 9 miles from Karen's house.
Step-by-step explanation:
Let the distance covered by Josh be [tex]S_{1}[/tex] and the distance covered by Karen be [tex]S_{2}[/tex]
The distance between Josh and Karen's houses is 20 miles. That is, the total distance both of them will cover is 20 miles.
Hence, [tex]S_{1} + S_{2} = 20 miles[/tex]
From the question, if Josh takes a 10 minute break after he starts while Karen rides the whole time, this means
If Karen rides for a total of [tex]t[/tex] mins, then Josh would spend [tex]t-10[/tex] mins for the journey
Also, Josh rides at an average speed of 15 miles per hour
and Karen rides at an average speed of 10 miles per hour
Now, for Josh
Distance = [tex]S_{1}[/tex]
Speed = 15 mile/hour (Covert to mile/minute); (NOTE: 1 hour = 60 mins)
Speed= 0.25 mile/min
Time spent = [tex]t - 10[/tex] mins
From
[tex]Speed =\frac{Distance}{Time}[/tex]
Distance = Speed × Time
Distance = 0.25 mile/min × (t-10) min
[tex]S_{1}[/tex] = 0.25(t-10) mile
[tex]S_{1}[/tex] = 0.25t - 2.5
t = ([tex]S_{1}[/tex] + 2.5) / 0.25 ....... (1)
For Karen,
Distance = [tex]S_{2}[/tex]
Speed = 10 mile/hour (Covert to mile/minute);
Speed= [tex]\frac{1}{6}[/tex] mile/min
Time spent = t
[tex]Speed =\frac{Distance}{Time}[/tex]
Distance = Speed × Time
Then, [tex]S_{2}[/tex] = [tex]\frac{1}{6}[/tex] mile/min × t min
[tex]S_{2}[/tex] = [tex]\frac{1}{6}t[/tex] mile
t = 6[tex]S_{2}[/tex] ......... (2)
Equating equations (1) and (2), we get
([tex]S_{1}[/tex] + 2.5) / 0.25 = 6[tex]S_{2}[/tex]
Then,
([tex]S_{1}[/tex] + 2.5) = 1.5[tex]S_{2}[/tex]
Recall that
[tex]S_{1} + S_{2} = 20 miles[/tex]
Then, [tex]S_{1} = 20 - S_{2}[/tex]
∴[tex]20 - S_{2}[/tex] + 2.5 = 1.5[tex]S_{2}[/tex]
2.5[tex]S_{2}[/tex] = 22.5
[tex]S_{2}[/tex] = 22.5 / 2.5
[tex]S_{2}[/tex] = 9 miles
This means Karen covers a total distance of 9 miles
Hence, they will meet at a distance 9 miles from Karen's house.
