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Answer: the rate of the cabin cruiser in calm water is 12 mph.
the rate of the current is 3 mph.
Step-by-step explanation:
Let x represent the rate of the cabin cruiser in calm water.
Let y represent the rate of the current.
A cabin cruiser traveling with the current went 45 miles in 3 h. This means that the total speed is (x + y) miles per hour.
Distance = speed × time
Distance travelled with the current is
45 = 3(x + y)
Dividing through by 3, it becomes
15 = x + y - - - - - - - - - - -1
Traveling against the current it took 5 hours to go the same distance. This means that the total speed is
(x - y) miles per hour.
Distance travelled against the current is
45 = 5(x + y)
Dividing through by 5, it becomes
9 = x - y - - - - - - - - - - -2
Adding equation 1 to equation 2, it becomes
24 = 2x
x = 24/2
x = 12 mph
Substituting x = 12 into equation 1, it becomes
15 = 12 + y
y = 15 - 12
y = 3 mph
Rate of the cabin cruiser will be 12 miles per hour and the rate of the current is 3 miles per hour.
Let the speed of the current = r miles per hour
And the speed of the cabin cruiser in the calm water = c miles per hour
Speed of the cabin cruiser against the current = (c - r) miles per hour
Speed of the cabin cruiser with the current = (c + r) miles per hour
Since, expression for the speed is given by,
Speed = [tex]\frac{\text{Distance}}{\text{Time}}[/tex]
Therefore, speed of the cabin cruiser with the current = [tex]\frac{45}{3}[/tex] = 15 miles per hour
So the equation for the speed will be,
c + r = 15 ------ (1)
Time taken to cover 45 miles against the current = 5 hours
Therefore, equation for this situation will be,
c - r = [tex]\frac{45}{5}[/tex]
c - r = 9 ------- (2)
Now solve this system of equations,
Add equation (1) and (2),
(c + r) + (c - r) = 15 + 9
2c = 24
c = 12 miles per hour
Substitute the value of 'c' in equation (1),
12 - r = 9
r = 12 - 9
r = 3 miles per hour
Therefore, speed of the cabin cruiser will be 12 miles per hour and the rate of the current is 3 miles per hour.
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