Jared’s cycling speed through the foothills was

Question

Jared is training for a cross country cycling race. He rode 56 miles in 2 hours. After the first 32 miles through the foothills, he increased his speed by 12 miles per hour as he rode on flat roads for the last 24 miles. Find Jared’s cycling speed through the foothills. 

Answer 1

Total time 2 hours

32 miles at speed r for time t

24 miles at speed r+12 for time 2-t

 

distance = rate*time

 

rt = 32

(r+12)(2-t) = 24

 

from 1st equation  t = 32/r

 

(r+12)(2-32/r) = 24

2r - 32 + 24 - 384/r = 24

2r - 32 - 384/r = 0

2r² - 32r - 384 = 0

r² - 16r - 192 = 0

 

Factors of -192 that add to -16 are -24(8)

 

(r-24)(r+8) = 0

r = 24 or -8

Discarding the negative, his cycling rate is 24 mph

 

t = 32/r = 32/24 = 4/3 = 1 1/3 hours

= 1 hour 20 minutes at 24 mph for 1st 32 miles

 

2-t = 2/3 hours or 40 minutes at 36 mph for 24 miles