SOLUTION: Raffaele ran 8 miles and then walked 6 miles. If he ran 5 mph faster than he walked and the total time was 2 hours, then how fast did he walk?

Algebra ->  Customizable Word Problem Solvers  -> Travel -> SOLUTION: Raffaele ran 8 miles and then walked 6 miles. If he ran 5 mph faster than he walked and the total time was 2 hours, then how fast did he walk?       Log On

Ad: Over 600 Algebra Word Problems at edhelper.com


    

Question 207393This question is from textbook Elementary and Intermediate Algebra
: Raffaele ran 8 miles and then walked 6 miles. If he ran 5 mph faster than he walked and the total time was 2 hours, then how fast did he walk? This question is from textbook Elementary and Intermediate Algebra

Found 2 solutions by josmiceli, MathTherapy:
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
For the 8 miles running:
d%5Br%5D+=+r%5Br%5D+%2At%5Br%5D
For the 6 miles walking:
d%5Bw%5D+=+r%5Bw%5D%2At%5Bw%5D
given:
d%5Br%5D+=+8
d%5Bw%5D+=+6
t%5Br%5D+%2B+t%5Bw%5D+=+2
r%5Br%5D+=+r%5Bw%5D+%2B+5
---------------------
d%5Br%5D+=+r%5Br%5D+%2At%5Br%5D
8+=+%28r%5Bw%5D+%2B+5%29+%2At%5Br%5D
8+=+%28r%5Bw%5D+%2B+5%29%282+-+t%5Bw%5D%29
8+=+2r%5Bw%5D+%2B+10+-+r%5Bw%5D%2At%5Bw%5D+-+5t%5Bw%5D
and
d%5Bw%5D+=+r%5Bw%5D%2At%5Bw%5D
6+=+r%5Bw%5D%2At%5Bw%5D
therefore
8+=+2r%5Bw%5D+%2B+10+-+6+-+5t%5Bw%5D
8+=+2r%5Bw%5D+%2B+4+-+5t%5Bw%5D
2r%5Bw%5D+=+4+%2B+5t%5Bw%5D
r%5Bw%5D+=+2+%2B+%285%2F2%29%2At%5Bw%5D
substituting:
6+=+r%5Bw%5D%2At%5Bw%5D
6+=+%282+%2B+%285%2F2%29%2At%5Bw%5D%29%2At%5Bw%5D
6+=+2t%5Bw%5D+%2B+%285%2F2%29%2A%28t%5Bw%5D%29%5E2
12+=+4t%5Bw%5D+%2B+5%2A%28t%5Bw%5D%29%5E2
5%2A%28t%5Bw%5D%29%5E2+%2B+4t%5Bw%5D+-+12+=+0
solve using quadratic formula
t%5Bw%5D+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
a+=+5
b+=+4
c+=+-12
t%5Bw%5D+=+%28-4+%2B-+sqrt%28+4%5E2-4%2A5%2A%28-12%29+%29%29%2F%282%2A5%29+
t%5Bw%5D+=+%28-4+%2B-+sqrt%28+16+%2B+240+%29%29%2F10+
t%5Bw%5D+=+%28-4+%2B-+sqrt%28+256+%29%29%2F10+
t%5Bw%5D+=+%28-4+%2B-+16%29%2F10+
t%5Bw%5D+=+12%2F10
t%5Bw%5D+=+1.2 hrs
t%5Br%5D+=+2+-+1.2
t%5Br%5D+=+.8 hrs
d%5Bw%5D+=+r%5Bw%5D%2At%5Bw%5D
6+=+r%5Bw%5D%2A1.2
r%5Bw%5D+=+6%2F1.2
r%5Bw%5D+=+5
His rate walking was 5 mi/hr
check answer:
d%5Br%5D+=+r%5Br%5D+%2At%5Br%5D
8+=+%28r%5Bw%5D+%2B+5%29+%2At%5Br%5D
8+=+%28r%5Bw%5D+%2B+5%29%282+-+t%5Bw%5D%29
8+=+%285+%2B+5%29%282+-+1.2%29
8+=+10%2A.8
8+=+8
OK

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!

Raffaele ran 8 miles and then walked 6 miles. If he ran 5 mph faster than he walked and the total time was 2 hours, then how fast did he walk?

Let Raffaele's walking rate of speed be S

Then his running rate of speed is S + 5 (he ran 5 mph faster than he walked)

Time+=+distance%2Frate

Since he ran 8 miles, walked 6 miles, and did them in 2 hours, then
8%2F%28S+%2B+5%29+%2B+6%2FS+=+2
8(S) + 6(S + 5) = 2(S)(S + 5) [Multiplying equation by LCD, S(S + 5)]
8S+%2B+30+%2B+6S+=+2S%5E2+%2B+10S
2S%5E2+-+4S+-+30+=+0
2%28S%5E2+-+2S+-+15%29=+0
S%5E2+-+2S+-+15+=+0 Eliminate 2 = 0, as 2+%3C%3E+0

(S - 5)(S + 3) = 0

S = 5, or - 3 (Eliminate S = -3, as rate of speed CANNOT be negative)

Therefore, Raffaele's walking rate of speed = highlight_green%285%29 mph