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**Hint:**Whenever the boat is traveling upstream the then equivalent speed of the boat will be calculated by taking the difference of the speed of the boat and speed of the stream and if it is the case of upstream then the equivalent speed of the boat will be the sum of the speed of boat and stream. The formula to calculate time is ${\text{Time}} = \dfrac{{{\text{Distance}}}}{{{\text{Speed}}}}$.

**Complete step by step answer:**

Let, the speed of the stream be x.

It is given that the speed of the boat in the still water is 10 mph.

When the boat is traveling in the upstream, then it has to travel against the stream, so the speed of the boat will be the difference of boat in still water and speed of the stream, i.e., $10 - x$.

Similarly, when the boat is traveling in the downstream, then it has to travel with the stream, so the speed of the boat will be the sum of the speed of the boat in still water and speed of the stream, i.e., $10 + x$.

As, ${\text{Time}} = \dfrac{{{\text{Distance}}}}{{{\text{Speed}}}}$.

So, the time taken for the boat to travel upstream will be equal to $\dfrac{{36}}{{10 - x}}$.

Similarly, the time taken for the boat to travel downstream will be equal to $\dfrac{{36}}{{10 + x}}$.

It is given the time taken by the boat to travel upstream is 90 minutes more than to travel downstream.

Now, convert 90 minutes in hours by dividing it by 60, as in one hour there are 60 minutes,

so $\dfrac{{90}}{{60}} = \dfrac{3}{2}$ hours.

As the difference between time traveled by boat during upstream and downstream is $\dfrac{3}{2}$ hours.

Therefore,

$\dfrac{{36}}{{10 - x}} - \dfrac{{36}}{{10 + x}} = \dfrac{3}{2}$, now find the value of x by solving this equation.

$

\dfrac{{36}}{{10 - x}} - \dfrac{{36}}{{10 + x}} = \dfrac{3}{2} \\

\dfrac{{36\left( {10 + x} \right) - 36\left( {10 - x} \right)}}{{\left( {10 - x} \right)\left( {10 + x} \right)}} = \dfrac{3}{2} \\

\dfrac{{360 + 36x - 360 + 36x}}{{{{10}^2} - {x^2}}} = \dfrac{3}{2} \\

\dfrac{{72x}}{{100 - {x^2}}} = \dfrac{3}{2} \\

144x = 300 - 3{x^2} \\

3{x^2} + 144x - 300 = 0 \\

{x^2} + 48x - 100 = 0 \\

$

Now , factorize the equation ${x^2} + 48x - 100 = 0$ and find the value of x.

$

{x^2} + 48x - 100 = 0 \\

{x^2} + 50x - 2x - 100 = 0 \\

x\left( {x + 50} \right) - 2\left( {x + 50} \right) = 0 \\

\left( {x - 2} \right)\left( {x + 50} \right) = 0 \\

x = - 50,2 \\

$

As the value x cannot be negative because if we substitute this value to calculate the time taken for the boat to travel downstream will be negative and time cannot be negative.

So, the speed of the stream is 2 mph.

Therefore, option (A) is the correct answer.

**Note:**

This is the case of stream and boat, let the speed of the stream be some value x and use the formula of time to equate both the time taken by the boat, as it is mentioned in the question that it takes 90 minutes of more time to travel upstream than downstream, using this and formula of time the speed of stream can be calculated.

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