Answer:
The speed of the tug boat in still water is 7.5 mph
Explanation:
Firstly, we assign a variable to represent the speed of the tug boat in still water.
Let the speed of the tug boat in still water be x mph
Mathematically;
time = distance/speed
We calculate the time taken for the boat to travel upstream as follows;
Upstream distance = 24 miles
Upstream speed= speed of tugboat - speed of current = x-3
So the time taken to travel upstream is thus;
24/(x-3)
Let’s do same for downstream travel;
downstream distance = 28 miles
Downstream speed = Speed of tugboat + speed of current = x + 3
So the downstream travel time = 28/(x + 3)
Now from the question, the total travel time is 8 hours
So if we add the time taken to travel downstream plus the time taken to travel upstream, the total of both = 8 hours
Thus Mathematically;
24/(x-3) + 28/(x + 3) = 8
So let’s calculate x
24(x + 3) + 28(x-3) = 8(x + 3)(x-3)
= 24x + 72 + 28x - 84 = 8(x^2-9)
= 52x + 72-84 = 8x^2 -72
8x^2 -72-72+84-52x = 0
8x^2 - 60 -52x = 0
Let’s divide through by 4
2x^2 -13x -15 = 0
2x^2+ 2x -15x -15 = 0
2x(x + 1) -15(x + 1) = 0
(x + 1)(2x-15) = 0
x + 1 = 0 or 2x -15 = 0
x = -1 or 2x = 15
x = -1 or x = 15/2
x = -1 or 7.5
since speed cannot take a negative value, we ignore the narrative value and this means that the speed of the boat in still water is 7.5 mph