Friday, July 23rd, 2010

### Solution:

First, let’s plot f(x) =

Here’s what the graph looks like:

Now, the definition of “domain of a function” as follows: The domain of a given function is the set of “input” values for which the function is defined.” For instance, the domain of cosine would be all real numbers, while the domain of the square root would be only numbers greater than or equal to 0 (ignoring complex numbers in both bases).

As you can see from the above graph, the function is valid for all “real” values of x, except for 0. The set of “real numbers”, as you may know, can be thought of as points on an infinitely long number line. The shorthand way of saying “all real numbers except for x” is as follows:

Friday, July 23rd, 2010

### Solution:

First, you have to have a clear idea what y = 2x^2+4x-5 means. This is an equation that tells you that, for any value of x, you can find y by dividing by multiplying x by x, and then multiplying that by two, adding 2 times x and finally subtracting 5. Does that sound familiar?

Now, what is the table you have to make? You will list several values of x (which you will be using as x-coordinates in your graph), and for each one use the equation to find what y is. We’ll show the work:

For x = 1, **y** = 2*1*1 + 4*1 – 5 = 1

For x = 2, **y** = 2*2*2 + 4*2 – 5 = 11

For x = 3, **y** = 2*3*3 + 4*3 – 5 = 25

For x = 4, **y** = 2*4*4 + 4*4 – 5 = 43

For x = 5, **y** = 2*5*5 + 4*5 – 5 = 65

Do you see what we did there? We just replaced x in the equation by each value of x, and did the multiplication and addition.

Now your table will look like this:

x | y

–+—–

1 | 1

2 | 11

3 | 25

4 | 43

5 | 65

We’ve only shown the x,y values for 1 through 5, out there are an infinite number of x values that you could plug in, including negative numbers and fractions. You can use the resulting T chart to help you create a graph, which will look like this:

Friday, July 23rd, 2010

### Solution:

The total area is 40π=2πrh+2πr^2

=2πr*8 + 2πr²

=16πr + 2πr²

=16r + 2r²

If we rearrange the terms and divide the terms by 2 (in order to simplify the equation), we get:

r^{2} + 8r – 20 = 0

(r+10) (r-2) = 0

r = 2 or -10

Therefore, **r = 2** (assuming that -10 is not an option!).

Friday, July 23rd, 2010

### Solution:

Let’s assume that the sides of the box are vertical. In that case, the volume of the container would be the area of the base times the height. Since we know the volume is 720 cu. in., and the height is 40 =n, then we know that the area of the base is 18 sq. in. = 18*40=720).

We don’t have enough information to find the lengths of the sides of the container’s triangular base. There are many shapes of triangles whose area is equal to 18 sq. inches. We would need to know the lengths of at least two sides of the base of the container, plus the angle formed by these two sides. We can then use the cosine rule (see below) to figure out the required length of the third side.

**The “cosine rule”**

In the triangle below, the three sides have lengths a, b and c. Angle A is opposite side a; similarly for B and C.

The cosine rule is as follows: a^{2}=b^{2}+c^{2}– 2bc (cos(A))

This allows us to work out the required length of the third side of the triangle if we know the length of two sides and the angle between them.

Thursday, July 22nd, 2010

### Solution:

Here’s how you solve this type of question:

- Assume that the maximum number of birds he can purchase’s 200, since he does not have space for more than that. Of course, he could purchase more and then release them into the wild, but we don’t think that’s what your teacher has in mind!
- One option is to buy no birds at all, and keep the $250.
- Since chickens cost only $1 each, he could buy 200 chickens and have $50 left over.
- Or he could buy 125 ducklings and have no money left over.
- So there are lots of possibilities. The general formula would be given by
**c + 2d 250 and c + d 200**, where c is the number of chickens and d is the number of ducklings. Since we are dealing with a real world situation, we can also assume that c and d are non-negative integers. The solution space would look like this:

The above chart shows all the possible numbers of chickens and ducklings that would satisfy the constraints of space and money. As you can see, there are thousands of options.

Monday, July 19th, 2010

### Solution:

First recognize that this is in the form of (x^{2}-a^{2}), which can always be factored as (x+a)(x-a). If you multiply these two terms, you get x^{2}-ax+ax-a^{2}, which is equal to x^{2}-a^{2}.

So in this case, we can see that **a=3**,

so x^{2}-9

= x^{2}-3^{2}

= **(x+3)(x-3)**

Monday, July 19th, 2010

### Solution:

A function is a type of relation, i.e. it relates one set of numbers to another. The function (call it “f”) takes any member of one set (call it “x”) and connects it to a unique member of another set. Symbolically, we write this unique member of the other set as f(x). You can say “f of x” when you see this rotation.

Now, we can write the function as a rule or equation. For example, **f(x) = 2x**. In this example, each value of x as assigned double its value by the function f.