Average speed

Question

Meg rowed her boat upstream a distance of 75 mi and then rowed back to the starting point. The total time of the trip was 20 hours. If the rate of the current was 5 mph, find the average age speed of the boat relative to the water

Answer 1

Distance = (Rate)(Time), so Time = (Distance/(Rate)

 

Let x = speed of boat in still water

 

Speed downstream = x+5

Speed upstream = x-5

 

(Time upstream) + (Time downstream) = 20

 

75/(x-5) + 75/(x+5) = 20

 

75(x+5) + 75(x-5) = 20(x-5)(x+5)

 

                  150x = 20(x²-25)

 

20x² - 150x - 500 = 0

 

  2x² - 15x - 50 = 0

 

(2x +5)(x-10) = 0     x = -2.5 or x = 10

 

Since x can't be negative, x = 10 mph.

 

Answer 2

Boat speed x
current speed 5

against current x-5 mph
with current x+ 5 mph

Distance= 75 miles

Time against + time with = 20 hours

 

75 /( x + 5 ) + 75 /(x - 5 ) = 20

LCD = (x - 5 ) ( x +5 )

75 *( x - 5 ) + 75 (x + 5 ) = 20

75 x - 375 + 75 x + 375 = 20( x ² - 25 )

 

150x =20 x ² - 500

 

20 x ² - 150 x - 500=0

dividing equation with10

 

2x² -15x -50 =0

 

Finding the roots of the equation by quadratic formula

x=10 and x=-2.5

Ignore negative value
Speed of boat is 10 mph

Answer 3

75 = (r-5)*t --> t = 75/(r-5)

75 = (r+5)*(20-t)

 

 

 

75 = (r+5)(20 - (75/(r-5))

 

75(r-5) = (r+5) [ 20(r-5) - 75]  <--- multiplies both sides by r-5

 

75r - 375 = (r+5) [ 20r - 100 - 75 ]

 

75r - 375 = (r + 5) [ 20r - 175]

 

75r - 375 = 20r^2 - 175r + 100r - 875

 

0 = 20r^2 - 75r - 875 - 75r + 375

 

0 = 20r^2 - 150r - 500

 

0 = 2r^2 - 15r - 50

 

0 - ( 2r    +   5)(r    - 10    )

 

2r + 5 = 0 --> r = -5/2 <--- negative numbers not allowed

 

r-10 = 0 ---> r = 10

 

Going upstream (against the current), the speed is 10 mph - 5 mph = 5 mph. 15 hours needed to travel 75 miles.

Going downstream (with the current), the speed is 10 mph + 5 mph = 15 mph. 5 hours needed to travel 75 miles.

The total time is 20 hours, as given. The answer checks and is now proven to be correct.

 

The speed relative to the current is 10 mph.