Suppose a Word is a String of 8 letters of the Alphabet with Repeated Letters Allowed: (5 points, 1 point each) Show all work. Do not Answer with just a Number. 1. How many words are there? 2. How many words end with the letter N? 3. How many words begin with R and end with N? 4. How many words start with A or B? 5. How many words begin with A or end with B

Solution:

1. If you have 26 choices for each letter in a word, then there are 26 such one-letter words, 26² two-letter words, 26³ three-letter words, etc. There would be 26⁸ eight-letter words.
2. There would be 26⁷ eight-letter words that end with the letter N, which is the same as the number of 7 letter words (just add “N” to the end of each seven-letter word.
3. There would be 26⁶ eight-letter words that begin with R and end with N (just add R to the front and N to the end of all six-letter words.
4. There would be 2*26⁷ eight-letter words that begin with an A or B (add A to the beginning of every seven-letter word, then add B to the beginning of every seven-letter word).
5. There would be 2*26⁷ eight-letter words that begin with A or end with B (add A to the beginning of every seven-letter word, then add B to the end of every 7 letter word).

Complete These Ordered Pairs: (0, ) (2, ) (-1, ) y=3x-1 (-1, ) (0, ) (1, ) 2x+y=3 (0, ) (-1, ) (1, ) y=2x+1

Solution:

Each ordered paid represents x and y: (x,y)

To calculate y, you can plug in x.

If y=3x-1, then the given values of x will result in the pairs (0,-1) (2,5) (-1,-4).

If y=3-2x, then the given values of x will result in the pairs (-1,5) (0,3) (1,1).

If y=2x+1, then the given values of x will result in the pairs (0,1) (-1,-1) (1,3).

N-3/8=6

Solution:

If Utility Function is Expressed as U(x,y) = x^0.5 y^0.5 What is the Marginal Utility at Point (64,25) and (49,36)? Treat Y as a Constant. Would the Answer be MUx(x,y) = y. MUx(64,25) = 5 ? and MUx(49,36) = 6?

Solution:

MUx(x,y) =     Derivative of U(x,y), treating y constant.

MUx(64,25) =

MUx(49,36) =

Is 288/1155 in Lowest Terms

Solution:

1. First, factorize the numerator using prime numbers: 288 = 2*12*12 = 2*3*4*3*4=2*2*2*2*2*3*
2. Second, factorize the denominator: 1155 = 5*231=5*7*33 = 5*7*3*11
3. Notice that the numerator and denominator have only one factor in common (3).
4. Restate the ratio after removing the common factor
   1. New numerator = 2*2*2*2*2*3 = 96
   2. New denominator = 5*7*11 = 385
   3. Same ratio in lowest terms: 

What Experiment Have 3 Types of Variables?

Solution:

A scientific experiment has three types of variables: independent, dependent and controlled. Please click on this link to get details: http://www.sciencebuddies.org/mentoring/project_variables.shtml.

Factor These: 2w^2-5w-10

Solution:

We can factor the expression  as follows:  

To figure this out, we used the quadratic formula (see http://en.wikipedia.org/wiki/Quadratic_equation):

w = 

We recognized that y=  is a quadratic equation in the form of y=  where a=2, b=-5 and c=-10.

To confirm that our calculation are right, we decided to plot the graph of y = by plugging in various values of w.

As you can see, there is a root at approximately -1.3 and another root at approximately +3.8. So we know approximately what the values of a and b are. We could use a calculator to get closer and closer to the exact values of a and b. For example, if we try w=3.8, we get a value of y = -0.12. If we try a slightly different value, we can see if we’re getting closer to zero or farther away: w=3.81 produces a value of y=-0.018, so we know that 3.81 is a better approximation than 3.8. If we keep calculating closer approximations of the root, we will arrive at the value of b that we calculated using the quadratic formula:

Use Mathematical Induction to Prove That the Statements are true for Every Positive Integer n. 1*3+2*4+3*5+⋯+n(n+2)=n(n+1)(2n+7)/6

Solution:

First, let’s make sure we understand what we mean by “mathematical induction”: “Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. It is done by proving that the first statement in the infinite sequence of statements is true, and then proving that if any one statement in the infinite sequence of statements is true, then so is the next one.” (Wikipedia) We have pasted below another more detailed explanation of how to create a proof using mathematical induction (see “Appendix” below).

Let’s now take the equation you provided: 1*3+2*4+3*5+…+n(n+2)=

We can show that this is true for n=1:         1*3 = = = 3

Now let us assume that the statement is true for n = k. If it is, then we will prove that it has to be true for n=k+1:

WTS è =       +

=    + 

= 

=

=

=

=       QED

In the above proof, WTS means “want to show” and QED means “quod erat demonstratum” (“which was to be demonstrated”).

Appendix

Here’s another more detailed explanation of how to create a proof using mathematical induction:

A Jeweler Needs to mix an Alloy with 16% Gold Content & an Alloy with a 28% Gold Content to Obtain 32oz. of a new Alloy with a 25% Gold Content. How Many oz. of Each of the Original Alloys Must be Used?

Solution:

• A = number of ounces of the first alloy
• B = number of ounces of the second all
• A+B=32
• 0.16 A +0.28 B = 32 x 0.25
• Since A + B = 32, we know that A = 32 – B. You can then re-write the equation above as 0.16 (32-B) + 0.28 B = 32 x 0.25
• This can be simplified to 5.12- 0.16 B + 0.28 B =8
• Or, 0.12 B = 2.88
• Or B = 24
• So A = 8

Solve the Triangle. Round Angle Measures to the Nearest Minute & Side Measures to the Nearest Tenth. a=11.4, b=13.7, c=12.2

Solution:

These are the formulas used to solve triangles:

1. The sum of the internal angles equals 180^o …A + B + C = 180^o

2. The ‘sine rule’ …

3. The ‘cosine rule’ …

a² = b² + c² – 2bc cosA

or

b² = a² + c² – 2ac cosB

or

c² = b² + a² – 2ba cosC

In the problem that you gave us, we know the length of the three sides of the triangle:

a = 11.4

b = 13.7

c = 12.2

When no angles are known, the cosine rule is the only option, so 11.4² =13.7² +12.2² – 2*13.7*12.2*cosA….Therefore cosA = 0.617955.

You can now use a calculator or a table to find the value of A. You should get A = 0.9047 radians = 51.833 degrees = 51 degrees and 50 minutes.

Use the sine rule to find one of the remaining angles.

=

So sin(B) =  = 0.944835

You can now use a calculator or a table to find the value of B. You should get B = 1.237 radians = 70.8801 degrees = 70 degrees and 53 minutes.

Finding the third angle is easy, since we know that A + B +C =180 degrees. C = 180 – 51.833 – 70.880 = 57.287 degrees = 57 degrees and 17 minutes.

Visual representation: