the current in a stream moves at a speed of 2 mph. A boat travels 8mi upstream and 8mi downstream in a total time of 7 hr. What is the speed of the boat in still water

T = 8/(V - 2) + 8/(V + 2) = 7

Solve for V

To find the speed of the boat in still water, let's denote the speed of the boat as "b" and the speed of the stream as "s".

When the boat is moving upstream (against the current), its effective speed is reduced by the speed of the stream. So the boat's speed relative to the ground is "b - s".

When the boat is moving downstream (with the current), the speed of the stream helps the boat move faster. So the boat's speed relative to the ground is "b + s".

Now we can use the given information to set up two equations:

Equation 1: The time taken to travel 8 miles upstream is equal to the distance divided by the effective speed: 8 / (b - s) = t₁

Equation 2: The time taken to travel 8 miles downstream is equal to the distance divided by the effective speed: 8 / (b + s) = t₂

We are given that the total time taken for the round trip is 7 hours, so we can create another equation:

Equation 3: t₁ + t₂ = 7

Now let's solve the equations to find the values of b and s.

From Equation 1, we can express t₁ in terms of b and s: t₁ = 8 / (b - s)

Similarly, from Equation 2, we can express t₂ in terms of b and s: t₂ = 8 / (b + s)

Substituting these values in Equation 3, we get: 8 / (b - s) + 8 / (b + s) = 7

Now, to avoid dealing with fractions, let's get rid of the denominators by multiplying both sides of the equation by (b - s)(b + s):

8(b + s) + 8(b - s) = 7(b - s)(b + s)

Expanding this equation gives us: 8b + 8s + 8b - 8s = 7(b² - s²)

Simplifying further: 16b = 7b² - 7s²

Rearranging the equation to form a quadratic equation in terms of b:

7b² - 16b - 7s² = 0

Now we have a quadratic equation in terms of b. We can solve this equation using the quadratic formula:

b = (-(-16) ± sqrt((-16)² - 4(7)(-7s²))) / (2(7))

Simplifying further:

b = (16 ± sqrt(256 + 196s²)) / 14

Now, to determine the positive value of b (since speed cannot be negative), we can eliminate the negative solution:

b = (16 + sqrt(256 + 196s²)) / 14

Therefore, the speed of the boat in still water is given by (16 + sqrt(256 + 196s²)) / 14.