word problems? your favorite? my nightmare help !?

Question

a stream flows at a rate of 5mph. a boat travels 75mi downstream and returns in a total time of 8hr. what is the speed of the boat in still water?

Answer 1

distance / rate = time

Use "b" for the boat's speed. Then we know that the upstream speed (b-5) and downstream (b+5) took 8 hours together for 75 miles each way:

75/(b + 5) + 75/(b - 5) = 8

Multiply by (b-5)(b+5) to eliminate the denominators:

75(b-5) + 75(b+5) = 8(b^2 - 25)
150b = 8b^2 - 200
8b^2 - 150b - 200 = 0

Use the quadratic:

b = (150 +/- sqrt(150^2 - 4*8*(-200)))/(2*8)
b = (150 +/- 170)/16

b = 20 mph, b = -1.25 mph

The -1.25 mph answer can be ignored, as the boat can't travel at a negative maximum speed.

Testing the answer:

75/(20 + 5) + 75/(20 - 5) =
75/25 + 75/15 =
3 + 5 =
8

Answer 2

time = distance / speed

the time it takes the boat to travel upstream plus the time it takes the boat to travel downstream equals 8 hr

let b be the boat of the current. b + 5 is the resultant speed at which the boat travels downstream and b - 5 is the resultant speed at which the boat travels upstream

75/(b + 5) + 75/(b - 5) = 8

75(b-5) / (b^2 - 25) + 75(b+5) / (b^2 - 25) = 8

75b - 375 + 75b + 375 = 8b^2 - 200

150b = 8b^2 - 200

0 = 8b^2 - 150b - 200

use quadratic formula and you'll get b = 20 mi/r

Answer 3

Let the speed of the boat in still water be x mph
Hence the resultant speed of the boat in downstream is x+5 and in upstream is x-5 mph
So,according to the condition of the problem,
75/(x+5) +75/(x-5)=8
or,{75(x-5)+75(x+5)}=8
or,(75x-375+75x+375)/(x^2-25) =8
or150x/(x^2-25)=8
or,8x^2-200=150x {by cross-multiplication]
or8x^2-150x-200=0
or 4x^2-75x-100=0
or,4x^2-80x+5x-100=0
or,4x(x-20)+5(x-20)=0
or,(x-20)(4x+5)=0
rejecting the negative value of x,we get that x=20
Therefore the required speed of the boat is 20 mph

Answer 4

Equation to solve: distance= (velocity x time)
Let V be velocity in still water
Let V+5 be downstream velocity
Let V-5 be upstream velocity
Let td be downstream time
Let 8-td be upstream time.
Downstream (V+5)td = 75
Upstream (V-5)(8-td)=75
The only combination of V and td that solves both equations is V=20, the still-water speed and td=3.

Answer 5

x = speed of boat in still water
x+5 = speed of boat downstream
x-5 = speed of boat upstream
75/(x+5) + 75/(x-5) = 8
75x- 375 + 75x+375 = 8(x^2-25)
150 x= 8x^2 -200
8x^2-150x-200 = 0
(x-20)(8x+10) =0
x = 20 mph