Problem Solving and Solving Roots
Work Problems
Example
Suppose that George can finish his homework in 5 hours and Carmen can finish her homework in 4 hours. How many hours will it take for them to finish their homework if they do it together?
Solution
We can think of this problem as a distance-rate-time problem where the distance for the whole test is 1. To find George's rate we write:
d = rt
1 = (r)(5) or r = 1/5
To find Carmen's rate, write
1 = (r)(4) or r = 1/4
Hence the total rate is
George's rate + Carmen's rate = 1/5 + 1/4
This is the amount of homework done in 1 hour. If together it takes x hours to do the homework then they will complete 1/x of the homework in 1 hour.
1
1 1
+
=
5
4 x
1
1
1
(20x) + (20x)
=
(20x)
5
4
x
4x + 5x = 20
9x = 20
x = 20/9 = 2 2/9
It takes 2 2/9 hours for them to finish their homework together.
Example
A boat travels 10 mi/hr in still water. If the boat takes the same amount of time to travel 3 miles upstream as 2 miles downstream, find the speed of the current.
Let
x = speed of the current.
then the rate upstream is
Rate Upstream = 10 - x
and the rate downstream is
Rate Downstream = 10 + x
since
d = rt
t = d/r Divide both sides by r
For the two trips, the time is the same, so
3
2
=
10 +
x
10 - x
Cross multiply:
3(10 - x) = 2(10 + x)
30 - 3x = 20 + 2x
10 = 5x
x = 2.
The current is going at 2 miles per hour.
Finding Roots
|
Definition of the nth root of x We define the nth root of x
to be y if
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Examples
What is the solution to
y2 = -3 ?
Since the square of any number is positive, the above equation has no real solution. In general negative numbers do not have even roots.
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