Ratextime=distance

Question

Boat a leaves the dock traveling north at 13mph. Boat B leaves the dock one hour later traveling at 10mph. How long until they are 100 miles apart?

Answer 1

Boat A is traveling 13mph.

Boat B leaves 1 hour later at 10mph.

This means that in that hour, Boat A is 13m out.

So, at hour 1, distance = 13m.

At hour 1, Boat A will be another 13m out, and Boat B will be 10m out.

So, at hour 2, distance = 26m - 10m = 16m.

At hour 2, Boat A will be another 13m out, and Boat B will be 20m out.

So, at hour 3, distance = 39m - 20m = 19m.

We can rewrite what we know the following way:

At hour 1, distance = 1 * 13m - 0 * 10m

At hour 2, distance = 2 * 13m - 1 * 10m

At hour 3, distance = 3 * 13m - 2 * 10m

At hour x, distance = x * 13m - (x - 1) * 10m

Now that we have a formula, we can turn it into an equation to find when distance will be 100m:

100m = x * 13m - (x - 1) * 10m

Let's get rid of the m's so we're just looking at the equation:

100 = 13x - 10(x - 1)

Simplify out that last part in the parentheses:

100 = 13x - 10x + 10

Add like terms:

100 = 3x + 10

Subtract the 10 from each side:

90 = 3x

And lastly divide by 3:

30 = x

Our answer is 30 hours!

To test, let's plug this back in to our formula:

distance = 30 * 13 - (30 - 1) * 10

Observing PEMDAS, let's solve the parentheses:

distance = 30 * 13 - 29 * 10

Next comes multiplication:

distance = 390 - 290

And lastly our subtraction:

distance = 100

So it works! At 30h, distance is 100m.