Modeling and Density

Objective

• Review over Scale Factors
• Review over changes in Perimeter, Area, and Volume
• Review over how to undo operations
• Go over how to find the density of objects
• Go over how to find the density of populations
• Go over how to find the density of energy
• Do some examples
• Homework

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Finding the Perimeter of a Dilated shape

To start off, let’s look at a shape that has been dilated by a certain factor first.

Then we can look at shapes that have only parts of them dilated by a scale factor.

So, first, we need an original shape:

And now, let’s measure the sides:

Now we know that the perimeter of this shape is going to be:

3 + 3 + 3 + 3

= 12

But what if we dilated this shape by a scale factor.

Say, 3?

Well, then we would have:

And now, let’s measure the sides:

Now we know that the perimeter of this shape is going to be:

9 + 9 + 9 + 9

= 36

And taking the area of the bigger shape, divided by the smaller shape gives us:

So, the perimeter is going to be 3 times the original perimeter.

So now we know we can multiply the original perimeter by the scale factor to get the new perimeter.

But what if it’s not the whole thing?

Well, to start, we first need an object:

So, as we can see, the lengths of each side of our square is 3.

And since the perimeter is just the sum of the sides of the object

Then we know the perimeter will be:

3 + 3 + 3 + 3

= 12

But what about when we multiplied all of the x-coordinates by a factor of 3?

What happens to the perimeter?

Well, let’s look:

Again, we know the perimeter will be:

9 + 3 + 9 + 3

= 24

And that seems a little weird, but let’s look at what we did.

We added an extra 6 units to each side

But left 2 of the sides the same.

So, if we were to write that out, it would look something like:

12 + 2(9 – 3)

So, the formula that we are looking for, would be:

= 12 + 2(6)

= 12 + 12

= 24

Finding the Area of a Dilated shape

To start off, let’s look at a shape that has been dilated by a certain factor first.

Then we can look at shapes that have only parts of them dilated by a scale factor.

So, first, we need an original shape:

And now, let’s measure the sides:

Now we know that the area of this shape is going to be:

3 * 3

= 9

But what if we dilated this shape by a scale factor.

Say, 3?

Well, then we would have:

And now, let’s measure the sides:

Now we know that the area of this shape is going to be:

9 * 9

= 81

And taking the area of the bigger shape, divided by the smaller shape gives us:

But that doesn’t seem to work because our scale factor is 3.

But what is 9?

Isn’t it 3 squared?

So, to find the new area, we take the scale factor, square it, then multiply it by the original area!

But what if it’s not the whole thing that is dilated?

Well, to start, we first need an object:

So, as we can see, the lengths of each side of our square is 3.

And since the area is just the base times the height.

Then we know the area will be:

3 * 3

= 9

But what about when we multiplied all of the x-coordinates by a factor of 3?

What happens to the area?

Well, let’s look:

Again, we know the area will be:

9 * 3

= 27

And now, if we take our new area, and divide it by the original area, we get:

So, what this tells us is:

If the shape has only a part of it dilated by a scale factor

To find the new area

We multiply the old area by the scale factor.

Or:

FINDING THE VOLUME OF A DILATED 3-D SHAPE

For volume, we’re only going to look at shapes that are dilated throughout the entire shape.

Mainly because if we talk about only dilating a part of the shape, it can get really complicated.

So to start, we need a shape:

And now, let’s measure the sides:

Now we know that the volume of this shape is going to be:

5 * 2 * 3

= 30

But what if we dilated this shape by a scale factor.

Say, 4?

Well, then we would have:

And now, let’s measure the sides:

Now we know that the volume of this shape is going to be:

20 * 8 * 12

= 1920

And taking the volume of the bigger shape, divided by the smaller shape gives us:

But that doesn’t seem to work because our scale factor is 4.

But what is 64?

Isn’t it 4 cubed?

So, to find the new volume, we take the scale factor, cube it, then multiply it by the original volume

So now, let’s do some examples:

5

3

2

20

12

8

EXAMPLE 1:

Find the new perimeter of the shape if the x-coordinates of the shape are multiplied by 2.

So, we have a few ways of finding this out.

We can multiply each x-coordinate by 2

Then use the distance formula to find the new distance of each side

Then add all of the sides together

Or

We can use the formula:

So:

EXAMPLE 2:

Find the new area of the shape if the y-coordinates of the shape are multiplied by 6.

So, we have a few ways of finding this out.

We can multiply each x-coordinate by 6

Then use the distance formula to find the new distance of each side

Then find the area

Or

We can use the formula:

So:

EXAMPLE 3:

Find the new Volume of the shape if the shape is multiplied by 7.

So, for this one, we need to use the formula:

So first, let’s find the volume of this shape:

12

7.2

9.6

3

Now, we plug it in, and get:

So, how do we undo operations?

First we need to understand what all of the operations that we will be undoing are.

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So, the operations are:

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1. Addition
2. Subtraction
3. Multiplication
4. Division

So, how do we undo Addition?

By subtracting. Whenever you see a + sign and you want to undo it, you subtract whatever is being added.

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An example of this:

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0 = x + 23

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If we want to know what x is, we undo the addition.

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So we subtract 23 from the equation.

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-23 -23

And we get that x = -23

So how do we undo Subtraction?

With addition! Whenever you see a - sign and you want to undo it, you add whatever is being subtracted.

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An example of this:

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x – 20 = 35

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If we want to know what x is, we undo the subtraction.

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So we add 20 from the equation.

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+ 20 + 20

And we get that x = 55

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So what about multiplication?

We undo multiplication with division. However, sometimes multiplication can be hard to spot.

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So an example is this:

If we have something like: 2x = 40

2x is the same as saying 2 * x, so if we want x by itself we divide by 2.

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2 2

After we divide by 2, we have that x = 20

SO THEN HOW DO WE UNDO DIVISION?

Following the same logic, if undoing multiplication is using division, then undoing division must mean we multiply.

If we want to undo division, we need to multiply. The easiest number to multiply by is the denominator so we can get x by itself

5 * * 5

After we multiply the entire equation by 5 we get that x = 125

SO WHAT ABOUT WHEN YOU HAVE MULTIPLE PROCEDURES GOING ON AT ONCE?

WAIT SADMEP?

Yes, to solve for x, we want to work in reverse.

Now, we don’t always have to, but it’s usually the easiest way to solve.

Here’s our example from before:

So, following SADMEP, what we want to do first is undo the subtraction.

To do that, we need to:

And what we’ll get is:

Again, following SADMEP again, what we want to undo now is the division

To do that, we need to:

And finally, we are left with:

SO THAT WAS A LOT OF REVIEW

But it’s all stuff you need to know, mainly because we’re going to be doing a lot more algebra formulas in this lesson

With some geometry formulas as well.

So, now let’s talk about density:

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.