The boat travels 60 miles to an island and the same 60 miles coming back. Changes in the wind and tide made the average speed on the return trip 3 mph slower than the speed on the way out. If the total time of the trip took 9 hours, find the speed going to the island and the speed on the return trip.

return speed = v - 3

time out = 60/v
time back = 60/(v-3)
so
60/v + 60/(v-3) = 9
60(v-3) + 60 v = 9 v(v-3)

9 v^2 -27v = 120 v - 180

9 v^2 - 147 v + 180 = 0
3 v^2 - 49 v + 60 = 0
(3 v -4)(v-15) = 0
v = 15 or v = 4/3
4/3 will not work because v - 3 is negative
try v = 15
15 and 15-3 = 12
then
60/v = 4
60/12 = 5
sure enough 9 hours

To solve this problem, let's assign variables to the unknowns.

Let's say the speed on the way out is "x" miles per hour (mph).
Since the speed on the return trip is 3 mph slower than the speed on the way out, the speed on the return trip can be represented as "x - 3" mph.

Now, we can use the formula: Speed = Distance / Time to find the time for each leg of the trip.

On the way to the island, the boat has to travel 60 miles at a speed of "x" mph.
So, the time taken for this leg of the trip is: Time = 60 miles / x mph, which can also be written as 60/x hours.

Similarly, on the way back, the boat has to travel 60 miles at a speed of "x - 3" mph.
So, the time taken for this leg of the trip is: Time = 60 miles / (x - 3) mph, which can also be written as 60/(x - 3) hours.

According to the given information, the total time taken for the trip is 9 hours.
Therefore, the equation can be set up as:

60/x + 60/(x - 3) = 9

To solve this equation, we need to find the values of "x" that satisfy it.

One way to solve it is by multiplying every term of the equation by the least common multiple (LCM) of the denominators of the fractions, which in this case is "x(x - 3)".

By doing so, the equation becomes:

60(x - 3) + 60x = 9x(x - 3)

Now, simplify and rearrange the equation:

60x - 180 + 60x = 9x^2 - 27x

Combine like terms:

120x - 180 = 9x^2 - 27x

Rearrange the equation to set it equal to zero:

9x^2 - 147x + 180 = 0

Now, we can either factor or use the quadratic formula to solve this equation.

By factoring, it becomes:

(3x - 4)(3x - 45) = 0

Setting each factor equal to zero:

3x - 4 = 0 or 3x - 45 = 0

Solving for "x" in each equation:

3x = 4 or 3x = 45

x = 4/3 or x = 45/3

x = 4/3 or x = 15

Since the speed of the boat cannot be negative, we can disregard the solution x = 4/3.

Therefore, the speed on the way out is x = 15 mph, and the speed on the return trip is (x - 3) = 12 mph.

Let's assume the speed of the boat going to the island is "x" mph.

Since the boat travels 60 miles to the island, the total time for this leg of the trip can be calculated using the formula: time = distance / speed.

So, the time going to the island is 60 / x hours.

Given that the average speed on the return trip is 3 mph slower than the speed going to the island, the speed on the return trip would be (x - 3) mph.

Since the boat also travels 60 miles on the return trip, the time for this leg of the trip can be calculated as well: time = distance / speed.

So, the time coming back is 60 / (x - 3) hours.

According to the problem, the total time of the trip is 9 hours.

Therefore, the equation can be set up as: 60 / x + 60 / (x - 3) = 9.

To solve this equation, we need to find a common denominator and then simplify the equation.

Multiplying each term by x(x - 3), we get: 60(x - 3) + 60x = 9x(x - 3).

Expanding and simplifying the equation: 60x - 180 + 60x = 9x^2 - 27x.

Combining like terms: 120x - 180 = 9x^2 - 27x.

Rearranging the equation to set it to zero: 9x^2 - 147x + 180 = 0.

Now we can solve this quadratic equation to find the values of x (speed going to the island) and (x - 3) (speed on the return trip).

Using factoring, the equation can be rewritten as: (9x - 12)(x - 15) = 0.

Setting each factor to zero:
9x - 12 = 0 or x - 15 = 0.

Solving for x, we find that x = 12 or x = 15.

Since the speed cannot be negative, we discard the x = 12 solution.

Therefore, the speed of the boat going to the island is 15 mph, and the speed on the return trip is 15 - 3 = 12 mph.