Problem Solving with Constraints

Objective

• Review over density
• Review over the different types of density
• Review over how to solve for density
• Go over how to deal with problems with constraints
• Do some examples
• Homework

So what is density?

Well, according to Google:

“Density is the amount of matter that an object has in a given unit of volume.”

So, in other words, it’s how much something can hold

This goes back to the old question,

Which is bigger, a pound of feathers:

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Or a pound of lead?

Well, we know that lead is more dense than feathers

Which means we can hold more lead in a smaller volume

And since feathers is less dense than lead, then it takes up more volume

So, the pound of feathers would be bigger than the pound of lead.

(Notice, we’re talking about which is larger, not which weighs more.)

Which leads us to the formula for density:

Different Types of Density

So, as you may have guessed, there are different types of density

Mainly because it’s an English word we throw around a lot

We use it for measuring the strength of materials

We use it to determine whether an area is overpopulated

We even use it to measure how much energy we can get from a substance

And all of these have different formulas

But they all are very similar.

Now, we’ve talked about density as far as strength

(I.E. since lead is more dense than feathers, lead is a stronger substance)

Now let’s talk about:

Population Density

So, to figure out population density, we need to think about what it is.

We’re not really talking about volume so much as space.

Or area.

We’re talking about how many animals (or people) are in a certain area

So, to find the population density, we’re going to use the same formula,

Just manipulate it to fit our needs.

So, our formula will be:

But again, this isn’t the only type of density that we deal with.

The other kind that is most common is:

Measures of Energy

So the last type of density we will be talking about is the type that deals with energy.

Now, there are a couple of things we need to go over:

First, the standard unit of energy is a British thermal unit (BTU).

It’s how much energy is needed to increase the temperature of one pound of water, one degree.

Second, is how volume comes into it.

So, what we use (so far) to release energy is usually either gas or coal (mostly gas)

And to make sure we can control the energy we’re generating, we have to measure how many BTU’s a substance can generate

Given a certain volume.

Usually, the formula we use is:

We do this because, again, we want to be able to predict what will happen when we….well….light it.

So now that we’ve covered the different types of density

Let’s do some examples dealing with it:

EXAMPLE 1:

Determine which type of wood is denser:

So, to figure out which is denser, we need to use the formula:

So, let’s start out with the Aspen.

The first thing we need to find is the volume of the Aspen

So:

Now we can find its density:

Now let’s look at the Juniper.

Again, we need to find the volume.

So:

Now we can find its density:

So now we know the denser wood is Juniper

EXAMPLE 2:

Colorado has a population of 5,268,367. Its territory can be modeled by a rectangle approximately 280 miles by 380 miles. Find the approximate population density of Colorado

So, to figure out the density, we need to use the formula:

Now, we just start plugging in what we know:

So, in the state of Colorado, there’s about 50 people per square mile.

SO WHAT DOES IT MEAN TO HAVE A CONSTRAINT?

Well, first let’s define a constraint.

So, according to Google:

“A constraint is a limitation or restriction.”

Which is completely right!

Basically what is meant, is how real world problems have limitations

And how you need to really think logically to solve these problems.

This is one of those lessons that’s easier to show than explain:

CONSTRAINT EXAMPLE:

We also know that our volume is 150, so:

So let’s say you have 150 cubic inches of wax, and you want to make a candle.

You know that the best way to make a candle, is to make sure that the candle’s height is the same as its diameter.

So, what would the radius be to make this candle?

Well, to start, we need a cylinder to represent the candle:

Now, we know we want the height to be equal to the diameter

Which consequently is the same as two times the radius

Now what else do we know?

We know we have 150 cubic inches of wax

Cubic means volume

So, we can reverse engineer the volume formula and find the radius!

So:

And we know that the height is equal to 2 * r

And now we can solve:

And that’s how it’s done.

So, to be fair, solving problems with constraints is very similar to solving problems regularly.

The difference is, we’re working with limited information

So, we need to make sure we understand how to continue forward.

So, to help with that, here are some more examples:

Example 1:

Find the volume of the following box:

So as we can see, there’s a lot here that we don’t know.

So, let’s start with what we do know

We know that the shortest sides are 60 cm,

And that they are broken into 4 equal parts.

So:

Now that we know that x = 4,

We can plug it into the other side as well

So:

Now that we know what all of the sides are equal to

We can find the Volume of this box

So:

So, now we know the volume of this box is: 37,125 cubic centimeters

Example 2:

Since we just found the measurement of the radius

Now we can find the height as well!

So, we backwards engineer the height as:

The design for a coffee mug says it needs to be cylindrical, with a height of 1.5 times its diameter, and a capacity of 450 mL when filled to the top.

What radius and height should the coffee mug have when it’s done? (1mL = 1cm cubed)

So for this one, we need to think about what we have:

Now, we know that the height of the mug is 1.5 times the diameter.

And we know that the diameter is twice the radius

So:

h = 1.5 * d

d = 2 * r

h = 1.5 * (2 * r)

h = 3r

Now that we know what the height is

And we know what the volume is

We can find the measure of the radius by plugging it all into the volume formula!

So:

So, our radius is about 3.6 cm

And our height is about 10.8 cm!