Respuesta :
Answer:
Average speed of boat = 3.67 km/h
Average speed of stream = 0.63 km/h
Step-by-step explanation:
30 km upstream in 10 hours gives a speed of 30/10 = 3 km/h relative speed of boat and stream
Same way, 44 km in 10 hrs = 4.4 km/h relative speed of boat and stream.
Let the speed of the stream be y
Let the speed of both be x, then
x - y = 3.... (1)
x + y = 4.4.... (2)
Subtract 1 from 2
2y = 1.4
y = 1.4/2 = 0.7 km/h
substitute value of y in 1
x - 0.7 = 3
x = 3.7 km/h
Also, 40 km upstream in 13 hrs gives speed of 3.08 km/h
55 km in 13 hrs gives 4.2 km/h
Let speed of boat be x and that of stream be y
x - y = 3.08.... (1)
x + y = 4.2..... (2)
Subtract 1 from 2
2y = 1. 12
y = 0.56 km/h
Substitute value of y in 1
x - 0.56 = 3.08
x = 3.64 km/h
Average speed of boat = (3.7 + 3.64)/2
= 3.67 km/h.
Average speed of stream = (0.7 + 0.56)/2 = 0.63 km/h
Answer:
- Let the speed of boat in stream be x km/hr
- And the speed of boat in still water be y km/hr.
- For upstream = x - y
- For downstream = x + y
As we know that,
[tex]\bigstar \: \: \sf Time = \dfrac{Distance}{Speed} \\ \\ [/tex]
[tex]\bigstar\:\underline{\boldsymbol{According\: to \:the\: Question\:now :}} \\[tex]
[tex]:\implies \sf \dfrac{30}{x - y} + \dfrac{44}{x + y} = 10 \\ \\ \\ [/tex]
[tex]:\implies \sf \dfrac{40}{x - y} + \dfrac{55}{x + y} = 13 \\ \\ \\[/tex]
[tex]\sf Let \: \dfrac{1}{x - y} = \textsf{\textbf{m}} \sf \: \: and \: \: \sf{\dfrac{1}{x + y} = \textsf{\textbf{n} }}\\ \\ \\[/tex]
[tex]:\implies \sf 30m + 44n = 10\: \: \: \: \Bigg\lgroup \textsf{\textbf{Equation (i)}}\Bigg\rgroup \\ \\ \\[/tex]
[tex]:\implies \sf 40m + 55n = 13\: \: \: \: \Bigg\lgroup \textsf{\textbf{Equation (ii)}}\Bigg\rgroup \\ \\ \\[/tex]
[tex]\qquad\tiny \underline{\frak{ Multiply \: equation \: (ii) \: by \: 3 \: and \: equation \: (i) \: by \: 4 :}} \\[/tex]
[tex]:\implies \sf 120m + 176n = 40\: \: \: \: \Bigg\lgroup \textsf{\textbf{Equation (iii)}}\Bigg\rgroup \\ \\ \\[/tex]
[tex]:\implies \sf 120m + 165n = 39\: \: \: \: \Bigg\lgroup \textsf{\textbf{Equation (iv)}}\Bigg\rgroup \\ \\ \\[/tex]
[tex]\qquad\tiny \underline{\frak{ Substracting \: equation \: (iii) \: from \: equation \: (iv) \: we \: get :}} \\[/tex]
[tex]\sf 120m + 176n = 40 \\ \\
\sf 120m + 165n = 39 \\ \\ [/tex]
[tex]\sf \: \: ( - ) \:\:\: \: \: ( - ) \: \: \: \: \: \: \: ( - ) [/tex]
_____________________
[tex]\: \: \: \: \qquad\sf 11n= 1 \\[/tex]
[tex]\: \: \: \: \: \qquad\sf n= \dfrac{1}{11} \\ \\ [/tex]
[tex]\qquad\tiny {\frak{ Put\: n = \dfrac{1}{11}\:in\:equation \: (i) \: we \: get :}} \\[/tex]
[tex]\dashrightarrow\:\:\sf 30m + 44 \times \dfrac{1}{11} = 10 \\ \\ \\[/tex]
[tex]\dashrightarrow\:\:\sf 30m= 10 - 4 \\ \\ \\[/tex]
[tex]\dashrightarrow\:\:\sf 30m= 6 \\ \\ \\[/tex]
[tex]\dashrightarrow\:\:\sf m= \dfrac{6}{30} \\ \\ \\ [/tex]
[tex]\dashrightarrow\:\:\sf m= \dfrac{1}{5} \\ \\ \\[/tex]
____________________....
[tex]\dashrightarrow\:\:\sf \dfrac{1}{x - y} = m\\ \\ \\[/tex]
[tex]\dashrightarrow\:\:\sf x - y= 5\: \: \: \: \Bigg\lgroup \textsf{\textbf{Equation (v)}}\Bigg\rgroup \\ \\ \\[/tex]
[tex]\dashrightarrow\:\:\sf \dfrac{1}{x + y} = n\\ \\ \\[/tex]
[tex]\dashrightarrow\:\:\sf x + y= 11\: \: \: \: \Bigg\lgroup \textsf{\textbf{Equation (vi)}}\Bigg\rgroup \\ \\ \\[/tex]
[tex]\qquad\tiny {\frak{Adding\:equation \: (v) \: and \: equation \: (vi) \: we \: get :}} \\[/tex]
[tex]:\implies \sf 2x = 16 \\ \\ \\[/tex]
[tex]:\implies \underline{ \boxed{ \textsf {\textbf{x = 8 km/hr}}}} \\ \\[/tex]
[tex]\qquad\tiny {\frak{Putting\:x = 8\: in \: equation \: (v) \: we \: get :}} \\[/tex]
[tex]:\implies \sf 8 - y = 5 \\ \\ \\ [/tex]
[tex]:\implies \sf y = 8 - 5 \\ \\ \\[/tex]
[tex]:\implies \underline{ \boxed{ \textsf {\textbf{y = 3 km/hr}}}} \\ \\[/tex]
_________________....
[tex]\bigstar\:\underline{\sf{Therefore\: speed\: of \:boat\: in\: still \:water\: and\: speed\: of\: stream:}} \\[/tex]
[tex]\bullet\:\:\textsf{Speed of boat in stream = \textbf{8 km/hr}}\\[/tex]
[tex]\bullet\:\:\textsf{Speed of boat in still water = \textbf{3 km/hr}}\\[/tex]
