SOLUTION: A person rowed her boat upstream for a distance of 32 miles and then rowed backed to the starting point. The total time of the trip was 18 hours. If the rate of the current was 7

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Question 862389: A person rowed her boat upstream for a distance of 32 miles and then rowed backed to the starting point. The total time of the trip was 18 hours. If the rate of the current was 7 mph, find the average of the boat in still water.
My daughter and I are stuck on this problem, many thanks.
Found 3 solutions by Alan3354, josmiceli, richwmiller:
Answer by Alan3354(69443)   (Show Source): You can put this solution on YOUR website!
A person rowed her boat upstream for a distance of 32 mil and then rowed backed to the starting point. The total time of the trip was 18 hours. If the rate of the current was 7 mph, find the average of the boat in still water.
---------
d = distance one way
c = speed of the current = 7
b = boat's speed
----
d = r*t
t = d/r
18 = 32/(b-7) + 32/(b+7)
18*(b-7)*(b+7) = 32*(b+7) + 32*(b-7)
9*(b^2 - 49) = 32b

Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation (in our case ) has the following solutons:



For these solutions to exist, the discriminant should not be a negative number.

First, we need to compute the discriminant : .

Discriminant d=2788 is greater than zero. That means that there are two solutions: .




Quadratic expression can be factored:

Again, the answer is: 4.71119528498757, -1.15563972943202. Here's your graph:

======================================
b =~ 4.7112 mi/hr
------------------------------
It's not a simple problem.
Answer by josmiceli(19441)   (Show Source): You can put this solution on YOUR website!
Let = the time in hours to row upstream
= the time in hours to row downstream
Let = the rate of the boat in still water in mi/hr
------------------------------------------------
Rowing upstream:
(1)
Rowing downstream:
(2)
----------------------------
(1)
and
(2)
(2)
(2)
(2)
(2)
(2)
(2)
(2)
Use the quadratic formula to solve, then
find
Check my math,I could have made an error
The method should be OK




Answer by richwmiller(17219)   (Show Source): You can put this solution on YOUR website!
Both of the other tutors started out fine but made errors.
There is a simple answer.
What grade is your daughter in?
18 = (64 b)/((b-7) (b+7))
b=9
the boat goes 9 mph
One started out fine but made some simple errors
18 = (64 b)/(b^2-49)
18(b^2-49)-64b=0
18b^2-64b-49*18=0
18b^2-64b-882=0
using the quadratic equation
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation (in our case ) has the following solutons:



For these solutions to exist, the discriminant should not be a negative number.

First, we need to compute the discriminant : .

Discriminant d=67600 is greater than zero. That means that there are two solutions: .




Quadratic expression can be factored:

Again, the answer is: 9, -5.44444444444444. Here's your graph:


and factoring
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Start with the given expression.



Factor out the GCF .



Now let's try to factor the inner expression



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



Looking at the expression , we can see that the first coefficient is , the second coefficient is , and the last term is .



Now multiply the first coefficient by the last term to get .



Now the question is: what two whole numbers multiply to (the previous product) and add to the second coefficient ?



To find these two numbers, we need to list all of the factors of (the previous product).



Factors of :

1,3,7,9,21,27,49,63,81,147,189,441,567,1323,3969

-1,-3,-7,-9,-21,-27,-49,-63,-81,-147,-189,-441,-567,-1323,-3969



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to .

1*(-3969) = -3969
3*(-1323) = -3969
7*(-567) = -3969
9*(-441) = -3969
21*(-189) = -3969
27*(-147) = -3969
49*(-81) = -3969
63*(-63) = -3969
(-1)*(3969) = -3969
(-3)*(1323) = -3969
(-7)*(567) = -3969
(-9)*(441) = -3969
(-21)*(189) = -3969
(-27)*(147) = -3969
(-49)*(81) = -3969
(-63)*(63) = -3969


Now let's add up each pair of factors to see if one pair adds to the middle coefficient :



First NumberSecond NumberSum
1-39691+(-3969)=-3968
3-13233+(-1323)=-1320
7-5677+(-567)=-560
9-4419+(-441)=-432
21-18921+(-189)=-168
27-14727+(-147)=-120
49-8149+(-81)=-32
63-6363+(-63)=0
-13969-1+3969=3968
-31323-3+1323=1320
-7567-7+567=560
-9441-9+441=432
-21189-21+189=168
-27147-27+147=120
-4981-49+81=32
-6363-63+63=0




From the table, we can see that the two numbers and add to (the middle coefficient).



So the two numbers and both multiply to and add to



Now replace the middle term with . Remember, and add to . So this shows us that .



Replace the second term with .



Group the terms into two pairs.



Factor out the GCF from the first group.



Factor out from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.



Combine like terms. Or factor out the common term



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



So then factors further to



===============================================================



Answer:



So completely factors to .



In other words, .



Note: you can check the answer by expanding to get or by graphing the original expression and the answer (the two graphs should be identical).
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