help solving speed formula

Question

Emily rows six miles downstream in 1 hour and her friend Amber is rowing 1 mile per hour faster, completes the trip in 2 hours.

 

1.  Find the spend of the current (c) and each girl's rowing speed

 

 

 

 

2.  If emily and amber were rowing spearately, who would complete their trip first and by how long?  rount to the

    nearest hundredth if necessary.

 

 

thanks

Answer 1

Hi again L;

 

distance=(rate)(time)

r=Emily's rate

c=rate-of-current

Emily...6 miles=(r+c)(1 hour)

Emily...(6 miles)/(1 hour)=r+c

Emily...6=r+c

Amber...6 miles=(r+1-c)(2 hours)

Amber...(6 miles)/(2 hours)=r+1-c

Amber...3=r+1-c

 

Emily.....6=r+c

Amber...3=r+1-c

Let's add the two equations together...

9=2r+1

8=2r

4=r

Emily's rate is 4 miles/hour.

Amber's rate is 4+1----5 miles/hour.

Let's subtract the second equation from the first...

3=-1+2c

4=2c

2=c

Current is 2 miles/hour.

 

Emily...6 miles=[(4 miles/hour)+(2 miles/hour)](1 hour)

Emily...6 miles=6 miles

Amber...6 miles=[(5 miles/hour)-(2 miles/hour)](2 hours)

Amber...6 miles=(3 miles/hour)(2 hours)

Amber...6 miles=6 miles

 

At 4+2 miles/hour, 6 miles/hour, Emily requires one hour to travel the distance of 6 miles.

At 4+1-2 miles/hour, 3 miles/hour, Amber requires two hours to travel the distance of 6 miles.

Answer 2

Hi L,

We think you meant that Amber was rowing 1 mph faster while rowing UPSTREAM.

1) The Distance they both row is presumably 6miles
2) Rate * Time = Distance
3) However Rate is the (Signed) sum of the rowers rates (Va,Ve) and the current rate (Vc)
4) Va=Ve+1 (Amber physically rows 1mph faster than Emily

5) The fact that Amber takes twice as long IMPLIES she is rowing AGAINST the current (but you didn't mention this).

PART 1:

(Ve+Vc)(1hr)=6miles
(Va-Vc)(2hr)=6miles

We can rewrite these equations by dividing by the times (1hr & 2hr respectively)

Ve+Vc = 6mph
Va-Vc = 3mph

We also know Ve=Va-1mph (substitute this into the first eqtn to obtain 2 eqtns with 2 unknowns.

(Va-1mph)+Vc = 6mph
Va-Vc = 3mph

Adding the two equations

2Va -1mph = 9mph (or 2Va = 10mph)....therefore...

Va=5mph
Ve=4mph (Recall Ve=Va-1mph)
Vc=2mph (Use ANY equation above to evaluate this)

PART 2 (This is also ambiguous) Your statement allows for Several interpretations

 

1) Both Amber & Emily row downstream

2) Both Amber & Emily row upstream

3) Emily rows downstream while Amber rows upstream (as originally posed)

 

CASE#1 ABOVE (Both Amber & Emily row downstream)

Obviously Amber will complete her trip first since she physically rows faster. Her trip will require the following time Ta

 

Ta=6miles/(Va+Vc)=6miles/(7mph)=6/7 hour

Te=1 hour (We already knew this)

Amber finishes FIRST by 1-(6/7)Hr=1/7hour (0.14 hrs to nearest hundredth)

 

CASE#2 ABOVE (Both Amber & Emily row upstream)

Ta=6miles/(Va-Vc)=6miles/(3mph)=2hr

Te=6miles/(Ve-Vc)=6miles/(2mph)=3hr

Amber finishes FIRST by 1hr

 

CASE#3 ABOVE (Emily downstream, Amber upstream)

 

We already know Emily beats Amber by an hour in the problem statement.  So they must not have been asking this question since it requires no additional work.

Answer 3

Speed(s) = Distance(d) / Time(t)

Now solving for Emily's speed call that (Es);

 

(Es) = her distance traveled (6miles) / Time traveled (1hr)

(Es) = 6miles / 1hr

(Es) = 6miles/hr

 

So Emily is traveling at a speed of 6miles per hour.

 

Now her friend Amber is traveling 1mile per hour faster than Emily.

In other words Amber's speed (As) would be Emily's speed (Es) + 1

 

Thus,

(As) = (Es) + 1

(As) = 6miles/hr + 1mile/hr

(As) = 7miles/hr

 

Now we know that Emily is traveling at a speed of 6miles per hour and Amber is traveling at a speed of 7miles per hour.

 

furthermore, We're told Amber completes this trip in 2hr.....lets assume her speed is constant/ remains the same throughout the entire trip.

 

What is the total distance traveled by Amber? we'd call that D

Remembering s = d/t; Amber is going at 7miles/hr and her time is 2hr

 

in respect to total distance, speed and time traveled;

S(Final Speed) = D(total distance) / Total time (T)

 

S = D/T

(7miles/hr) = D / 2hr

 

we'll cross multiply to solve and isolate D;

giving

 

D = (7miles/hr) * (2hr)

D = 14miles.

 

So the total distance is 14miles.

 

Find the Speed of the current(c)

giving the total distance of the course to be 14miles and the total time used to complete it was 2hr;

 

S(c) = D/T

S(c) = 14miles / 2hr

S(c) = 7miles/hr

 

Thus the speed of the current is 7miles/hr.

 

If Emily and Amber were rowing separately, who would finish first and by how long?

 

how long would it take Emily? here, we're solving for time.

Given Es = 6miles/hr and the total distance D = 14miles, time = ?

 

Using

S = D/T

cross multiplying both side

 

S*T = D

to solve and isolate T, we'd divide both sides by S, giving

 

T = D/S

where D = 14miles and S = 6miles/hr

 

T = 14miles / 6miles/hr

 

T = 2.3hr or 138mins

 

For Amber, we already know it took her 2hr to finish which is also 120mins

 

So Amber finishes in 2hr (120mins) and Emily in 2.3hr (138mins)

 

By How long?

Emily's time -  Amber's time

 

138mins - 120mins = 18mins

 

So Amber finishes 18mins before Emily.

-----------------------------------------------------------------------------------------------------------------------------

 

1. Find the speed of the current (c) and each girl's rowing speed

  - Speed of Current = 7miles/hr

 

  - Emily's Speed = 6miles/hr

 

  - Amber's Speed = 7miles/hr

2. If Emily and amber were rowing separately, who would complete their trip first and by how long? round to the nearest hundredth if necessary.

 

Amber finishes first at a speed of 7miles/hr in 2hrs

 

Emily finishes 18mins after Amber at a speed of 6miles/hr with a time of 2.3hr