A tugboat goes 24 miles upstream and 28 miles downstream in a total of 8 hours on a river that has a current of 3 mph. Find the speed of the tugboat in still water.

The speed of the tug boat in still water is 7.5 mph

Step-by-step 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

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