Question 818151: A cyclist traveled 60 mi at a constant rate before reducing the speed by 7 mph. Another 40 mi was traveled at the reduced speed. The total time for the 100-mile trip was 9 h. Find the rate during the first 60 mi.
Answer by TimothyLamb(4379)   (Show Source):
You can put this solution on YOUR website! s = d / t
d = s * t
---
total trip:
ta + tb = 9
tb = 9 - ta
s = 100 / 9 = 11.11 mph
---
segment a:
sa = 60 / ta
---
segment b:
sb = 40 / tb
sb = sa - 7
40 / tb = sa - 7
40 = tb(sa - 7)
40 = (9 - ta)(sa - 7)
40 = 9sa - 63 - sa*ta + 7ta
---
sa = 60 / ta
sa*ta = 60
---
40 = 9sa - 63 - sa*ta + 7ta
40 = 9(60/ta) - 63 - 60 + 7ta
40 = 540/ta - 123 + 7ta
163 = 540/ta + 7ta
163 = 540/ta + 7ta^2/ta
163ta = 540 + 7ta^2
7ta^2 - 163ta + 540 = 0
---
the above quadratic equation is in standard form, with a=7, b=-163, and c=540
---
to solve the quadratic equation, by using the quadratic formula, plug this:
7 -163 540
into this: https://sooeet.com/math/quadratic-equation-solver.php
---
the roots of the quadratic are:
ta = 19.2857143
ta = 4
---
the large root (19.3 hours) exceeds the total time for the trip (9 hours), so use the small root (4 hours):
---
answer:
ta = 4 hours
tb = 5 hours
sa = 60 / 4 = 15 mph
sb = 40 / 5 = 8 mph
---
check:
sa - sb = 7 mph (correct!)
---
Solve and graph linear equations:
https://sooeet.com/math/linear-equation-solver.php
---
Solve quadratic equations, quadratic formula:
https://sooeet.com/math/quadratic-formula-solver.php
---
Convert fractions, decimals, and percents:
https://sooeet.com/math/fraction-decimal-percent.php
---
Calculate and graph the linear regression of any data set:
https://sooeet.com/math/linear-regression.php
|
|
|
