Respuesta :
Answer:
Rowing rate of the guide in calm water is 6 mph.
Step-by-step explanation:
Let the rowing rate of the guide is x mph in the calm water.
Rate of river's current = 4 mph
Therefore, speed of the boat upstream = (x - 4) mph
and speed of the river downstream = (x + 4) mph
Time taken to row 5 miles upstream = [tex]\frac{\text{Distance traveled}}{\text{speed}}[/tex]
= [tex]\frac{5}{(x-4)}[/tex] hours
Time taken to row 5 miles downstream = [tex]\frac{5}{(x+4)}[/tex] hours
Since total time spent to row down and come back is = 3 hours
So [tex]\frac{5}{(x-4)}+\frac{5}{(x+4)}=3[/tex]
[tex]5[\frac{x+4+x-4}{(x-4)(x+4)}]=3[/tex]
5(2x) = 3(x - 4)(x + 4)
10x = 3(x² - 16)
3x² - 10x - 48 = 0
From quadratic formula,
x = [tex]\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}[/tex]
From our equation,
a = 3, b = -10 and c = -48
Now we plug in these values in the formula,
x = [tex]\frac{10\pm \sqrt{(-10)^{2}-4(3)(-48)}}{2(3)}[/tex]
= [tex]\frac{10\pm \sqrt{100+576} }{6}[/tex]
= [tex]\frac{10\pm \sqrt{676}}{6}[/tex]
= [tex]\frac{10\pm 26}{6}[/tex]
= 6, -2.67 mph
Since speed can not be negative so x = 6 mph will be the answer.
Therefore, rowing rate of the guide in calm water is 6 mph.
