# Math Help - Argh! Not story problems!!!

1. ## Argh! Not story problems!!!

I hate having to ask two questions in a row, but this story problem is killing me. I've got the answers to the first two parts of the question (I think) but the last part is the one I'm having difficulties with.

Lauren is competing in her town's cross-country competition. Early in the event, she must race from point A on the Greenbriar River to point B, which is 5 mi downstream and on the opposite bank. The Greenbriar is 1 mi wide. In planning her strategy, Lauren knows she can use any combination of running and swimming. She can run 10 mph and swim 8 mph. 1) How long would it take if she ran 5 mi downstream and then swam across? 2) Find the time it would take if she swam diagonally from A to B. 3) Find x so that she could run x miles along the bank, swim diagonally to B, and complete the race in 36 min. (Ignore the current in the river)

Anyone care to help me out with this one? It would be very much appreciated!!!

2. Originally Posted by potato92
I hate having to ask two questions in a row, but this story problem is killing me. I've got the answers to the first two parts of the question (I think) but the last part is the one I'm having difficulties with.

Lauren is competing in her town's cross-country competition. Early in the event, she must race from point A on the Greenbriar River to point B, which is 5 mi downstream and on the opposite bank. The Greenbriar is 1 mi wide. In planning her strategy, Lauren knows she can use any combination of running and swimming. She can run 10 mph and swim 8 mph. 1) How long would it take if she ran 5 mi downstream and then swam across? 2) Find the time it would take if she swam diagonally from A to B. 3) Find x so that she could run x miles along the bank, swim diagonally to B, and complete the race in 36 min. (Ignore the current in the river)

Anyone care to help me out with this one? It would be very much appreciated!!!
I assume that you mean point B is directly opposite a point 5 mi down the river on the same side as A. I am going to use B' to designate that point. Let x be the distance she runs down the river to point C. The distance remaining to point B' is 5- x and the distance from from B' to B is 1 mi. B, B', and C form a right triangle with BC the hypotenuse. By the Pythagorean theorem, its length, the distance from C to B, is $\sqrt{(5-x)^2+ 1}= \sqrt{x^2- 10x+ 26}$.

speed= distance/time so time= distance/speed. Since she can run 10 mph, she can run distance x in $\frac{10}{x}= 10x^{-1}$ hours. Since she can swim 8 mph, she can swim distance $\sqrt{x^2- 10x+ 26}$ in $\frac{8}{\sqrt{x^2- 10x+ 26}= 8(x^2- 10x+ 26)^{-1/2}$ hours.

That is, if she decides to run down the river x miles and then swim to point B, she will take a total of $10 x^{-1}+ 8(x^2- 10x+ 26)^{-1/2}$.

That is the function you want to minimize.

3. Hello, potato92!

Lauren is competing in her town's cross-country competition.
She must race from point $A$ on the Greenbriar River to point $B$,
which is 5 mi downstream and on the opposite bank. The Greenbriar is 1 mi wide.

Lauren knows she can use any combination of running and swimming.
She can run 10 mph and swim 8 mph.

1) How long would it take if she ran 5 mi downstream and then swam across?
Code:
      A - - - - - 5 - - - - - C
----o-----------------------o----
|
|
| 1
|
|
----------------------------o----
B

She would run 5 miles at 10 mph.
. . This would take: . $\tfrac{5}{10} \,=\,\tfrac{1}{2}$ hours.

She would swim 1 mile at 8 mph.
. . This would take: . $\tfrac{1}{8}$ hours.

Her total time: . $\frac{1}{2} + \frac{1}{8} \:=\:\frac{5}{8}$ hours.

2) Find the time it would take if she swam diagonally from A to B.
Code:
      A - - - - - 5 - - - - - C
----o-----------------------o----
*                   |
*               |
*           | 1
*       |
*   |
----------------------------o----
B

She would swim: . $\sqrt{5^2+1^2} \:=\:\sqrt{26}$ miles at 8 mph.

Her time is: . $\frac{\sqrt{26}}{8} \:\approx\:0.637$ hours.

3) Find $x$ so that she could run $x$ miles along the bank,
swim diagonally to B, and complete the race in 36 min.
Code:
      : - - - - - 5 - - - - - :
A - - -  x  - - - P 5-x C
----o-----------------o-----o----
\    |
\   |
\  | 1
\ |
\|
----------------------------o----
n  B

She would run $x$ miles to $P$ at 10 mph.
. . This takes her: . $\frac{x}{10}$ hours.

She would swim $\sqrt{(5-x)^2 + 1^2} \:=\:\sqrt{x^2 - 10x + 26}$ miles at 8 mpn.

. . This takes her: . $\frac{\sqrt{x^2-10x + 26}}{8}$ hours.

$\text{Her total time is: }\: 36\text{ minutes} \:=\:\tfrac{3}{5} \text{ hours.}$

. . $\frac{x}{10} + \frac{\sqrt{x^2-10x+26}}{8} \;=\;\frac{3}{5}$

. . And solve for $x.$

$\bigg[\,\text{There are }two\text{ possible answers: }\; x \;\approx\;0.641\,\text{ and }\,4.692\,\bigg]$