Caroline's Scribe Post

Thursday, November 29 and Friday, November 30!

To start off class, we had Unit 3 Quiz 2. Whew!! Whipping right through unit 3. 

OB had two problems to start class off written on the board about......OPTIMIZATION!

What a lovely word...so optimistic and cheery! The dictionary definition means to make the best or most effective use of a situation, opportunity, or resource. Something to think of as we live our lives!

But in mathematical terms, it means to rearrange or rewrite data to improve efficiency of retrieval or processing. We will learn how to optimize our calculations to get the answer in the most effective way!

So lets get started...

Consider two numbers whose product is 50. If their sum is as large as possible what are the two numbers?

O'Brien's strategy: mathemetize it!!

We shall: let x be one #.

                let y be another #.

What we know: sum = x+y

                          x*y = 50

What we need to do: Create a secondary equation to help solve the primary equation. Bring you back to Algebra with Seibert anyone..?

okay SO: y = 50/x

Substitute that into the primary equation.

Here is the working:

Next problem!!

Page 231/ #5

Inscribe a rectangle in an isosceles right triangle whose hypotenuse is 2 units long. What are the dimensions of the triangle with the largest area?

A beautiful diagram:

(kinda)

What we know:

You can find an equation of the line which will give you a value for y. Then, plug this value for y into the primary area equation. We know that y = (-x+1) because If you split the triangle in half, you will have two right triangles. The base of each separate triangle is 1, because the base of the entire triangle is 2. We know the height of the triangle is one because that is the rules of isosceles right triangles. Therefore, using Pythagorean theorem, you know that y = (-x+1). 

Here is the working to find the area:

Then we looked at #47
The trough in the figure is to be made to the dimensions shown. Only the angle θ can be varied. What value of θ will maximize the trough's volume?

If θ can vary, what will maximize the trough's volume?

We know the formula for volume is V=A*h.

If you split apart the trapezoid, you can make the base of each triangle on the end equal to sinθ, and the height of the triangle to be cosθ. Therefore, the top of the entire trapezoid would have a length of 1+2sinθ. 

If we know that, we can find the volume by multiplying the area times the height. The height is cosθ, the length is 20, and the base is 2+2sinθ.

To end class, Mr. O'Brien told us there were strategys for solving Max-Min Problems on page 223. They say this:

If you like word problems, you will like tonights IW.
IW #6: pg 231/ 5, 9, 13, 17, 20, 22, 31, 41, 47, 53, 55, 56


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Sorry for not having the scribe post up for next class.

To start off the 40 minute period class, we looked at a few problems from the text book from section 5.4, Modeling and Optimization. These were on the IW from the previous night and were questions asked on the class google doc.

#17. Designing a Can
You are designing a 1000-cm3 right circular cylindrical can whose manufacture will take waste into account. There is no waste in cutting the aluminum for the side, but the top and bottom of radius r will be cut from squares that measure 2r unites on a side. The total amount of aluminum used up by the can will therefore be:

What is the ratio of h to r?

To start this off, we drew a picture!

To solve this, we know we have a radius, height, and area. We need to formulate a second equation which relates radius and height. 

Alex came up with the perfect equation: If we are given that the area is 1000 cm3, then 

Don't be afraid to check your answers in the back of the book. If you are wrong, don't give up!! Look back over what the question is asking you.

#22. Maximizing Volume

Find the dimensions of a right circular cylinder of maximum volume that can be inscribed in a sphere of radius 10 cm. What is the maximum volume?

Here is a diagram: (remember, diagrams can be very helpful because they can show you things that you did not see by just reading the problem.)

See that once we drew a diagram, you can see Pythagoras's triangle!! This sets us up to create two equations. 

The first one: 

And we can create a second one using Pythagorean Theorem to relate r and x.

When thinking about the physical situation, if the radius is smaller or bigger, it reaches its maximum and minimum points. As it does this, the cylinder would look like a flat pancake if the radius is at its min or a tall pencil if the radius is at its max. The most optimum volume is when the radius is in the middle!

And there you have it!

Next scribe is Eddie McCluskey. :)