A ship travels downstream for 36 miles and then makes the same return trip upstream. The trip downstream took 3/4 of an hour less then the trip upstream. If the rate of the current is 4mph find the rate of the ship in still water and the time it takes to complete each part of the trip.

Careless error on my part, that equation should have been

144(x + 4) - 144(x - 4) = 3(x^2 - 16)
144x + 576 - 144x + 576 = 3x^2 - 48
3x^2 = 1200
x^2 = 400
x = ± 20 , came out much nicer!

I had defined x to be speed of the boat in still water, see above
so 20mph is the speed of the boat in still water, one of your questions.

As to the time, we defined them as 36/(x+4) and 36/(x-4)
using my calculator , those turned out as 36/24 hrs and 36/16 hrs
which is 1.5 hrs and 2.25 hrs.
Note the difference in those times is 2.25-1.5 = .75 or 3/4 hour as needed.

(I should have checked my previous answer and that would have shown an error)

speed of ship in still water --- x miles

speed downstream = x+4
speed upstream = x-4

time downstream = 36/(x+4)
time upstream = 36/(x-4)

36/(x-4) - 36/(x+4) = 3/4
multiply by 4(x+4)(x-4)

144(x + 4) - 144(x - 4) = x^2 - 16
expand, arrange as a quadratic, and solve for x

I'm getting x=+-4sqrt73

So how do I get the rate of ship in still water and time it takes to complete each part of trip?

Did I get the wrong answer?

For x

No worries. Thanks Reiny!

To solve this problem, let's define the variables:

Let's call the rate of the ship in still water "S" (in mph) and the time it takes for the trip downstream "T_down" (in hours).

Given that the rate of the current is 4 mph, we can define the speed of the ship traveling downstream as "S + 4 mph" and upstream as "S - 4 mph".

Now, let's solve the problem step by step:

1. Calculate the time it takes for the trip downstream:
Distance = Rate × Time
36 miles = (S + 4) mph × T_down

2. Calculate the time it takes for the trip upstream:
Distance = Rate × Time
36 miles = (S - 4) mph × (T_down + 3/4) hours
Since we're given that the downstream trip took 3/4 of an hour less, we add 3/4 hour to the time for the upstream trip.

Now, we have two equations:

1. 36 = (S + 4)T_down ---> Equation 1
2. 36 = (S - 4)(T_down + 3/4) ---> Equation 2

Let's solve the equations to find the values of S (the rate of the ship in still water) and T_down (the time it takes for the trip downstream):

From Equation 2, let's simplify:
36 = (S - 4)(T_down + 3/4)
36 = ST_down - 4T_down + 3S - 12

Rearranging and combining like terms:
36 = ST_down + 3S - 4T_down - 12
36 = ST_down + 3S - 4T_down - 12

Now, substitute the value of T_down from Equation 1 into Equation 2:
36 = (S + 4)T_down + 3S - 4T_down - 12

Simplifying the equation further:
36 = ST_down + 4T_down + 3S - 4T_down - 12
36 = ST_down - T_down + 3S - 12
36 = (S + 4)T_down - T_down + 3S - 12
36 = 4T_down + 4S - T_down - 12

Now, let's combine like terms:
36 = 3S + 3T_down - 12

Rearranging and simplifying:
24 = 3S + 3T_down
8 = S + T_down

Now, we have two equations:
1. 36 = (S + 4)T_down
2. 8 = S + T_down

Solve the equations simultaneously:

From equation (2), we have:
T_down = 8 - S

Now, substitute T_down = 8 - S into equation (1):
36 = (S + 4)(8 - S)

Multiply the terms:
36 = 8S + 32 - S^2 - 4S

Rearrange and combine like terms:
S^2 + 12S - 68 = 0

Now, we can solve this quadratic equation using factoring, completing the square, or the quadratic formula. Factoring doesn't work in this case, so let's use the quadratic formula:

S = (-b ± √(b^2 - 4ac)) / 2a

where a = 1, b = 12, and c = -68

Substituting the values into the formula:
S = (-12 ± √(12^2 - 4(1)(-68))) / (2 * 1)
S = (-12 ± √(144 + 272)) / 2
S = (-12 ± √416) / 2
S = (-12 ± 20.396) / 2

Simplify further:
S = (-12 - 20.396) / 2 or S = (-12 + 20.396) / 2
S = (-32.396) / 2 or S = (8.396) / 2
S = -16.198 or S = 4.198

Since the speed of the ship cannot be negative, we disregard the negative value.

Therefore, the rate of the ship in still water is approximately 4.198 mph.

To find the time it takes to complete each part of the trip, substitute the value of S into Equation 1:

36 = (4.198 + 4)T_down
36 = 8.198T_down + 16.792

Rearranging and solving for T_down:
8.198T_down = 36 - 16.792
8.198T_down = 19.208
T_down ≈ 2.344 hours (approximately)

Now, we know that the time for the upstream trip is T_down + 3/4 hours:
T_upstream ≈ 2.344 + 0.75
T_upstream ≈ 3.094 hours (approximately)

Therefore, the ship takes approximately 2.344 hours for the downstream trip and approximately 3.094 hours for the upstream trip.