Respuesta :
Answer:
6 mph
Step-by-step explanation:
Let x mph be the rate of the boat. Going up the river boat has the rate x-2 mph (the current decreases the rate of the boat) and going down the river boat has the rate x+2 mph (the current increases the rate of the boat). Then it will take [tex]\dfrac{12}{x-2}[/tex] hours to go up the river and [tex]\dfrac{12}{x+2}[/tex] hours to go down the river. Thus,
[tex]\dfrac{12}{x-2}-\dfrac{12}{x+2}=1\dfrac{1}{2},\\ \\\dfrac{12x+24-12x+24}{(x-2)(x+2)}=\dfrac{3}{2},\\ \\96=3(x^2-4),\\ \\x^2-4=32,\\ \\x^2=36,\\ \\x=\pm6\ mph.[/tex]
The rate of the boat cannot be negative, hence, x=6 mph.
Answer:
The rate of his boat is:
6 mph
Step-by-step explanation:
It is given that:
Ben's boat will take 1 1/2 hours longer to go 12 miles up a river than to return.
Let u denote the speed of the boat in still water.
and v denote the speed of the current.
Then the speed of boat upstream= u-v km/h
and speed of boat downstream=u+v km/h
Let t denote the time taken by the boat downstream.
Then the time taken by boat upstream is: t+(3/2) hours
Distance traveled by boat each way is: 12 miles.
Hence, we have:
Speed of boat upstream is:
[tex]\dfrac{12}{t+\dfrac{3}{2}}[/tex]
i.e.
[tex]u-v=\dfrac{12}{t+\dfrac{3}{2}}----------(1)[/tex]
and
speed of boat downstream i.e.
[tex]u+v=\dfrac{12}{t}------------(2)[/tex]
On subtracting equation (1) from equation (2) we have:
[tex]2v=\dfrac{12}{t}-\dfrac{12}{t+\dfrac{3}{2}}[/tex]
Also, we are given :
[tex]v=2\ mph[/tex]
i.e.
[tex]2\times 2=\dfrac{12}{t}-\dfrac{12}{t+\dfrac{3}{2}}[/tex]
i.e.
[tex]4=\dfrac{12\times (t+\dfrac{3}{2})-12\times t}{t(t+\dfrac{3}{2}}\\\\i.e.\\\\4=\dfrac{12t+18-12t}{t(t+\dfrac{3}{2})}\\\\i.e.\\\\4(t(t+\dfrac{3}{2}))=18\\\\i.e.\\\\2(t(t+\dfrac{3}{2}))=9\\\\i.e.\\\\2t^2+3t=9\\\\i.e.\\\\2t^2+3t-9=0[/tex]
i.e.
[tex]2t^2+6t-3t-9=0\\\\i.e.\\\\2t(t+3)-3(t+3)=0\\\\i.e.\\\\(2t-3)(t+3)=0\\\\i.e.\\\\t=\dfrac{3}{2}\ or\ t=-3[/tex]
But t can't be negative.
Hence, we have:
[tex]t=\dfrac{3}{2}[/tex]
Hence, from equation (2) we have:
[tex]u+v=\dfrac{12}{\dfrac{3}{2}}\\\\i.e.\\\\u+v=\dfrac{12\times 2}{3}\\\\i.e.\\\\u+v=8\\\\i.e.\\\\u+2=8\\\\i.e.\\\\u=8-2\\\\i.e.\\\\u=6\ mph[/tex]
Hence, the answer is: 6 mph
