more math questionss! i have the answers i just dont understand!?

Question

1. the current of a river is 5 miles per hour. a boat travels between two points on the river. if it takes 2 hours to travel in one direction and 5/2 hours to travel in the other direction, what i the speed of the boat in still water?
answer: 45 miles per hour

2. rationalize the denominator
44x^4y / 16z^3
that whole fraction is under a square root sign
answer: x^2 (/11yz) / 2z^2

the / before the 11yz is a square root sign

3. solve for the variable
(w-4)= /(7w+2)
the 7w+2 is under a square root sign
answer: w=14
i dont understand why the answer isnt -1 also?

thank you!!!

Answer 1

Let x = speed of the boat in still water.
With the current, the boat speed is (x + 5) mi/h.
Against the current, the boat speed is (x - 5) mi/h.

distance = speed * time
With current:
                    d = (x + 5) * 2
Against current:
                    d = (x - 5) * 2.5
so
     (x + 5) * 2 = (x - 5) * 2.5
         2x + 10 = 2.5x - 12.5
              22.5 = 0.5x
                   x = 45 mi/h
------------------
Factor the radicand and take the square roots of the factors:
√[(44x⁴y)/(16z³)] = √(44x⁴y)/√(16z³)
                           = (√4√11√x⁴√y)/(√16√z³)
                           = (2√11x²√y)/(4z√z)
                           = (x²√11√y)/(2z√z)
multiply by √z/√z:
                           = (x²√11√y√z)/(2z√z√z)
                           = (x²√11√y√z)/(2zz)
                           = (x²√(11yz)/(2z²)
-----------------------
               (w - 4) = √(7w + 2)
Square both sides of equation:
               (w - 4)² = 7w + 2
      w² - 8w + 16 = 7w + 2
    w² - 15w + 14 = 0
   (w - 14)(w - 1) = 0  ⇒  w=14, w=1.
w=1 can be eliminated because
               (w - 4) = √(7w + 2)
               (1 - 4) = √(7*1 + 2)
                     - 3 ≠ 3

Answer 2

1. If the speed in still water is S then when traveling with
the current, your speed is S + 5, and when traveling against
it, it is S - 5. Lets call the distance traveled D. Since distance
is equal to speed time time we have two cases

D = 2 * (S + 5)

and

D = 5/2 * (S - 5)

or
2*(S+5) = 5/2* (S-5)

expanding things a bit gives

2S + 10 = 5S/2 - 25/2

gathering all the S's to one side gives

10 + 25/2 = 5S/2 - 2S

or 45/2 = S/2

or S = 45

3. W-4 = SQRT(7W+2)

square both sides

(W-4)^2 = 7W + 2

or
W^2 - 8W + 16 = 7W + 2

or W^2 - 15W + 14 = 0

this can be factored into
(W-14)(W-1) = 0

so it would seem like both 14 and 1 would work.

But when you try 1 in the original equation you would
have

-3 = SQRT(9)

The 1st step of squaring both sides messes things up because while

(-3)^2 = SQRT(9)^2

-3 is not equal to SQRT(9)

Answer 3

1, You should always STATE the equation you use in these word problems. Here, it is
distance= speed x time. Let X be the still-water speed. Going upstream, the speed is X-5. Going downstream, the speed is X+5. The distance each way, however, is the same.
So D= 5/2 (X-5) = 2 (X+5)
5/2 X - 12.5 = 2X +10
1/2 X = 22.5 and X=45 mph.

In problem 3, you square both sides, arrange terms in descending orders of powers of w, and solve the resulting quadratic. However, when you square the sides, you introduce a "spurious" solution. If w=-1, the term in the sqrt( ) would be -5, which is imaginary and you prove that -5 = 5i

Answer 4

Let b = boat speed
One way the speed will add to the current speed, the other way it will subtract, but it is the same distance both times, so

2(b+5) = (5/2)(b-5)
4(b+5) = 5(b-5)
4b + 20 = 5b - 25
b = 45

2. A great explanation online so I don't have to put the radical signs in:

http://www.helpalgebra.com/articles/rationalizedenominator.htm

3. someone else already explained no negatives under the radical

Answer 5

3) You can not find the SQUARE root of a negtive number. . . And if W=-1 then the right side of the equation would be: The square root of negative five.

The square root of a negative number just doesn't make sense. . . because a positive times a positive makes a posive answer, and a negative times a negative makes a positive answer. . .

Answer 6

for question 3

square both sides to get

w^2 -8w +16 =7w +2
simplify to w^2 -15w +14=0
factorise to (w-1)(w-14)
w= 1 or 14

The others you can do yourself