Mathematical Models 2

How to translate from English to Mathematics

The Real World

Most mathematics questions are neat and pure and simple.

But the real world is often "messy": there are facts that get in the way, the numbers don't always work out nicely, we have to convert answers to different units and so on.

On Mathematical Models we showed how to make and use some simple models. Here are some more.


Example: River Cruise

Example: a 3 hour river cruise goes 15 km upstream and then back again. The river has a current of 2 km an hour. What is the boat's speed and how long was the upstream journey?

river boat back-and-forth in 3 hours

There are two speeds to think about: the speed the boat makes in the water, and the speed relative to the land:

Because the river flows downstream at 2 km/h:

We can turn those speeds into times using:

time = distance / speed

(moving 8 km at 4 km/h takes 8/4 = 2 hours, right?)

And we know the total time is 3 hours:

total time = time upstream + time downstream = 3 hours

Put all that together:

total time = 15/(b-2) + 15/(b+2) = 3 hours

Now we use our algebra skills to solve for "b".

First, get rid of the fractions by multiply through by (b-2)(b+2):

3(b-2)(b+2) = 15(b+2) + 15(b-2)

Expand everything:

3(b2-4)= 15b+30 + 15b-30

Bring everything to the left and simplify:

3b2 - 30b -12 = 0

It is a Quadratic Equation! And solving it gives the solutions:

b = -0.39 or 10.39 (to 2 decimal places)

b =-0.39 makes no sense for this real world question, but b = 10.39 is just perfect!

Answer: Boat's Speed = 10.39 km/h (to 2 decimal places)

And so the upstream journey = 15 / (10.39-2) = 1.79 hours = 1 hour 47min

And the downstream journey = 15 / (10.39+2) = 1.21 hours = 1 hour 13min


Example: Mixing Solutions

You have been asked to provide 150 mL of hydrochloric acid at a 12% concentration


In the cabinet you find:

How many mL of each should you use?

total HCl = HCl in 10% solution + HCl in 25% solution

(0.12)(150) = (0.10)(a) + (0.25)(150-a)

Solve: 18 = 0.1a + 37.5 - 0.25a

"a"s to left, rest to right: 0.25a - 0.1a = 37.5-18

Simplify: 0.15a = 19.5

a = 19.5/0.15 = 130

Answer: 130 ml of 10% solution and 20 ml of 25% solution

Note: I have a "Mix Slider" that could also have been used to solve this