Hello, potato92!
Lauren is competing in her town's crosscountry competition.
She must race from point $\displaystyle A$ on the Greenbriar River to point $\displaystyle B$,
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
oo


 1


o
B
She would run 5 miles at 10 mph.
. . This would take: .$\displaystyle \tfrac{5}{10} \,=\,\tfrac{1}{2}$ hours.
She would swim 1 mile at 8 mph.
. . This would take: .$\displaystyle \tfrac{1}{8}$ hours.
Her total time: .$\displaystyle \frac{1}{2} + \frac{1}{8} \:=\:\frac{5}{8}$ hours.
2) Find the time it would take if she swam diagonally from A to B. Code:
A      5      C
oo
* 
* 
*  1
* 
* 
o
B
She would swim: .$\displaystyle \sqrt{5^2+1^2} \:=\:\sqrt{26}$ miles at 8 mph.
Her time is: .$\displaystyle \frac{\sqrt{26}}{8} \:\approx\:0.637$ hours.
3) Find $\displaystyle x$ so that she could run $\displaystyle x$ miles along the bank,
swim diagonally to B, and complete the race in 36 min. Code:
:      5      :
A    x    P 5x C
ooo
\ 
\ 
\  1
\ 
\
o
n B
She would run $\displaystyle x$ miles to $\displaystyle P$ at 10 mph.
. . This takes her: .$\displaystyle \frac{x}{10}$ hours.
She would swim $\displaystyle \sqrt{(5x)^2 + 1^2} \:=\:\sqrt{x^2  10x + 26}$ miles at 8 mpn.
. . This takes her: .$\displaystyle \frac{\sqrt{x^210x + 26}}{8}$ hours.
$\displaystyle \text{Her total time is: }\: 36\text{ minutes} \:=\:\tfrac{3}{5} \text{ hours.}$
. . $\displaystyle \frac{x}{10} + \frac{\sqrt{x^210x+26}}{8} \;=\;\frac{3}{5}$
. . And solve for $\displaystyle x.$
$\displaystyle \bigg[\,\text{There are }two\text{ possible answers: }\; x \;\approx\;0.641\,\text{ and }\,4.692\,\bigg]$