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Dennis went cross country skiing for 6 hours on Saturday. He skied 20 miles uphill and then 20 miles downhill, returning to his starting point. His uphill speed was 5 mph slower than his downhill speed. What was dennis's speed going uphill and his speed going downhill?

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Respuesta :

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.

raphealnwobi

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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