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Answer: Downhill:10mph Uphill:5mph
Step-by-step explanation:
We are looking for Dennis’s downhill speed.
Let
r=
Dennis’s downhill speed.
His uphill speed is
5
miles per hour slower.
Let
r−5=
Dennis’s uphill speed.
Enter the rates into the chart. The distance is the same in both directions,
20
miles.
Since
D=rt
, we solve for
t
and get
t=
D
r
.
We divide the distance by the rate in each row and place the expression in the time column.
Rate
×
Time
=
Distance
Downhill
r
20
r
20
Uphill
r−5
20
r−5
20
Write a word sentence about the time.
The total time traveled was
6
hours.
Translate the sentence to get the equation.
20
r
+
20
r−5
=6
Solve.
20(r−5)+20(r)
40r−100
0
0
0
=
=
=
=
=
6(r)(r−5)
6
r
2
−30r
6
r
2
−70r+100
2(3
r
2
−35r+50)
2(3r−5)(r−10)
Use the Zero Product Property.
(r−10)=0
r=10
(3r−5)=0
r=
5
3
The solution
5
3
is unreasonable because
5
3
−5=−
10
3
and his uphill speed cannot be negative. So, Dennis's downhill speed is
10
mph and his uphill speed is
10−5=5
mph.
Check. Is
10
mph a reasonable speed for biking downhill? Yes.
Downhill:
10 mph
5 mph⋅
20 miles
5 mph
=20 miles
Uphill:
10−5=5 mph
(10−5) mph⋅
20 miles
10−5 mph
=20 miles
The total time traveled was
6
hours.
Dennis’ downhill speed was
10
mph and his uphill speed was
5
mph.
Dennis uphill speed is 5 mph and his downhill speed is 10 mph
Speed is the ratio of distance to time. It is given by:
Speed = distance/time
Let a represent the speed uphill and t₁ the time spent, hence:
a = 20/t₁
t₁ = 20/a
Let b represent the speed downhill and t₂ the time spent, hence:
b = 20/t₂
t₂ = 20/b
The total time spent was 6 hours, hence:
t₁ + t₂ = 6
20/a + 20/b = 6
But uphill speed was 5 mph slower than his downhill speed, hence:
a = b - 5
20/(b-5) + 20/b = 6
6b² - 30b = 20b + 20b - 100
6b²-70b + 100 = 0
b = 10 mph
a = b - 5 = 10 - 5
a = 5 mph
Hence Dennis uphill speed is 5 mph and his downhill speed is 10 mph
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