What is the Largest Amount of Postage That Cannot be Made Using These Stamp Denominations: 17 Cents, 9 Cents & 5 Cents?

Solution:

We’re not sure we understand the question, because if you have enough stamps (of any denomination) you can cross any amount. However, you said that the total value (let’s call is “q”) has to be less than a million dollars (e.g. $999,999.99), so let’s work backwards.  $999,999.99 can be obtained as follows:

1. 5,882,350 stamps of 17 cent denomination
2. 1 stamp of 9 cent denomination
3. 8 stamps of 5 cent denomination

So we have shown the largest amount of postage that can be paid (below the million dollar limit) is $999,999.99.

Now let’s assume that the amount paid has to be exact, i.e. we cannot overpay. This is a lot more complicated, as to do this we have to consider all values of q, where q = 17x + 9y + 5z, and where x, y and z are positive integers.  We could write a computer program to check the value of q for all values of x, y and z, and then see which values are missing, and finally pick the highest number among the missing values. We might want to work backwards from $999,999.99 (since we know we can reach $999,999.99) and check to see if the a small amount is possible. For example, we know that we can remove one of the 8 stamps of 5 cent denomination, so that gets us to $999,999.94. We could remove another 5 cent stamp to get to $999,999.89, and then add a 9 cent stamp to get to $999,999.98 (x= 5,882,352; y=2; z=6). We can continue this exercise until we hit a number that we cannot create with any combination of values for x, z and z.

The easiest way to solve the problem is to recognize to work backwards to prove that we can calculate most values of q below $1,000,000:

1. We have defined q=17x+9y+5z
2. We know that q-1=17(x)+9(y+1)+5(z-2)
3. So let’s start with (x=0; y=0, z=20,000,000), whereby q=$1,000,000
4. Now increase y by 1 and decrease z by 2: q=9,999,999.99
5. Again increase y by 1 and decrease z by 2: q=9,999,999.98
6. Continue this until z reaches zero (x=0; y=10,000,000; z=0), where  q=$900,000
7. Start again with (x=0;y=0;z=18,000,000), where q=$900,000
8. Repeat the process of decreasing 1 y by one and decreasing z by 2 (you can do this in an Excel spreadsheet)
9. Continue until z reaches zero, and then reset y to zero and z to whatever is required to reach the last value of q (or a number slightly higher)
10. Repeat the exercise until you reach a value of q = 32 cents.
11. Now you need the 17 cent stamp to get to q=31 (x=1,y=1,z=1).
12. You will also need the 17 cent stamp for q=26 and q=22 (see blow)
13. You can find solutions for q down to 22, but you will get stuck on the number 21…so that must be the answer!

-2x < 3/7

Solution:

Cos 2x=0.32

Solution:

First let’s apply the reverse cosine function to both sides of the equation:

cos^-1(cos(2x)) = cos^-1 (0.32) = cos^-1 ()

Now if you take the inverse cosine the  cosine of 2x, you get 2x…so:

2x = cos^-1 ()

2x 71.34°

x 35.67°  (which is approximately the same as 0.622 radians)

This looks like it might be the final answer, but actually it’s only one of the many correct answers. One way to see this is by graphing y=cos(2x) and seeing where it intersects the line y=0.32. If you do this, you will see that the intersections occur at multiple points:

As you can see, the intersections occur at x 0.622 radians, 2.519 radians, 3.764 radians, 5.660 radians, etc. The general form of the solution set is as follows:

What is the Volume of a Sphere the Radius of Six?

Solution:

1. The formula for the volume of a sphere is 
2. r = 6
3. = 

What is an Inverse Function?

If you have a function called ƒ, let’s call its inverse ƒ^–1 (i.e. ƒ^–1 is the inverse function of ƒ).

By definition, the property of ƒ^–1 is that if ƒ(a)=b, then ƒ^–1(b)=a. Wikipedia (http://en.wikipedia.org/wiki/Inverse_function) has a good example:

A function ƒ and its inverse ƒ^–1. Because ƒ maps a to 3, the inverse ƒ^–1 maps 3 back to a. In mathematics, if ƒ is a function from a set A to a set B, then an inverse function for ƒ is a function from B to A, with the property that a round trip from A to B to A (or from B to A to B) returns each element of the initial set to itself.

Solve them by substitution method: 5b – 3a = 2, (a-1)^2 + (b-1)^2 = 34

Solution:

First, we can write  

Then we can substitute this into the second equation you get

We can simplify this to

If multiply both sides by 25 and expand this out we get

This can be simplified to: Or

Thus the solutions for a are 6 and -4

The solutions for b can be found by the equation (b=4 when a=6, and b=-2 then a=-4)

• Another way to solve this is by looking at the equations graphically. When you plot the two equations, you can see there they intersect.

As you can see, they intersect at two places:

You can plug the answers into both equations to confirm that they are both true using these two values of a and b.

Is this funtion linear? if so, rewrite in y=b+mx. g(w)= 1-12w/3

Solution:

• g(w)=1-12w/3

• g(w)=1 + (-4)w

• In the above example, y=g(w), b=1 and m=-4. If we replace each of the terms we can rewrite the above as y=b+mx

2x-5y=9 and 3x+4y=25. Solve by the method of elimination.

Solution:

1.    3x + 4y=25
2.    2x – 5y=9

We could rewrite these as follows by multiplying the first equation by 5 and the second by 4:

1.    15x + 20y = 125
2.    8x  – 20y = 36

If we add the above two equations, we can eliminate y:

23x = 161

So x =3D 7, and plugging this into the first equation we can figure out
y:  21 + 4y =3D 25,

so y = 1

Solve this problem by using elimination: 2x-3y=61 and 2x+y=-7

Solution:

1. 2x –  =y=61
2. 2x +   y= -7

If you subtract the second equation from the first, you can eliminate x and you are left with -4y = 68.

So y =3D -17, and plugging this into the first equation we can figure out

y:  2x = (-51)=61,

so x = 5

Little Nero is a lost neurotransmitter who just found himself atop a receptor of the receiving dendrite and needs to get his charge to the neighboring neuron…i need detailed instructions to Little Nero so he is able to reach a successful action potential??

Solution:

First some definitions:

A neuron is an electrically excitable cell that processes and transmits information by electrochemical signaling. On one end of the neuron are the dendrites (receiving end) and the other end is the transmitting end (axon terminal).

Synapse: One neuron connects to another neuron when the dendrite of one is connected to the axon terminal of the other. This connection is called a synapse.

Dendrites are the branched projections of a neuron that act to conduct the electrochemical stimulation received from other neural cells to the main part of the neuron from which the dendrites project. Dendrites form the main receiving part of neurons. Dendrites collect and funnel these signals to the soma and axon.

A neurotransmitter is a chemical that is released from a neuron.

Synaptic Cleft: The tiny space between two nerve cells across which the neurotransmitter diffuses

Receptor: Neurotransmitters cross the synapse where they may be accepted by the next neuron at a specialized site on a dendrite called a receptor.

Here are the detailed instructions for the “lost” neurotransmitter:

• You are currently on the receptor of a dendrite

• To get to the next neuron in the chain, you must travel the length of the neuron you are current on

• Travel across the dendrite

• Travel across the soma

• Travel down the entire length of the axon

• Travel across the axon terminal

• At the synaptic cleft, bind with receptor sites on the neighboring neuron’s dendrite