It takes Nemo 2 hours more to swim 15 miles upstream than to go back downstream. If it takes Nemo to swim at 4 miles per hour in still water, what's the rate of the current?

To solve this problem, let's assume the speed of the current is represented by 'c' miles per hour.

When swimming upstream, Nemo is swimming against the current, so the effective speed will be the speed in still water (4 miles per hour) minus the speed of the current (c miles per hour).

When swimming downstream, Nemo is aided by the current, so the effective speed will be the speed in still water (4 miles per hour) plus the speed of the current (c miles per hour).

Let's use the formula: distance = speed × time to solve for the time it takes Nemo to swim upstream and downstream.

For upstream:
distance = 15 miles
speed = 4 miles per hour - c miles per hour (as Nemo is swimming against the current)
time = t hours (unknown)

15 = (4 - c) * t

For downstream:
distance = 15 miles
speed = 4 miles per hour + c miles per hour (as Nemo is swimming with the current)
time = t - 2 hours (as it takes Nemo 2 hours less to swim downstream)

15 = (4 + c) * (t - 2)

Now we have a system of two equations with two unknowns, t and c. Let's solve this system.

From the first equation:
15 = (4 - c) * t
Divide both sides by (4 - c):
15 / (4 - c) = t

From the second equation:
15 = (4 + c) * (t - 2)
Simplify:
15 = (4 + c) * t - (4 + c) * 2
15 = (4 + c) * t - 8 - 2c
15 + 8 = (4 + c) * t - 2c
23 = (4 + c) * t - 2c

Substitute t with 15 / (4 - c) from the first equation:
23 = (4 + c) * (15 / (4 - c)) - 2c

Next, simplify and solve for c.

23 = (4 + c) * (15 / (4 - c)) - 2c
Multiply both sides by (4 - c):
23 * (4 - c) = (4 + c) * 15 - 2c * (4 - c)
92 - 23c = 60 + 15c - 8c + 2c^2
Rearrange and simplify:
2c^2 + 31c + 60 - 92 = 0
2c^2 + 31c - 32 = 0

Now we can solve this quadratic equation to find the value(s) of c. We can use factoring, completing the square, or the quadratic formula.

Using the quadratic formula: c = (-b ± √(b^2 - 4ac)) / 2a
For this equation, a = 2, b = 31, and c = -32.

c = (-(31) ± √((31)^2 - 4 * 2 * (-32))) / 2 * 2
c = (-31 ± √(961 + 256)) / 4
c = (-31 ± √(1217)) / 4

Calculating the value using a calculator, we get two possible rates for the current:

c ≈ 1.96 (rounded to two decimal places) or c ≈ -16.46 (rounded to two decimal places).

Since the rate of the current cannot be negative (as it represents the speed of water flow), the rate of the current is approximately 1.96 miles per hour.