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Math Help - can anybody help me on this problems...

  1. #1
    rcs
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    can anybody help me on this problems...

    i keep trying on these problems it can really get the correct answers. even if there are given... these are some of the problems i kind of confused to solve. even if the answers are there but i need help to reach on the solutions how they are done.

    1. Solve for (x; y) in the system (e^x + 2)^2 - y = 3, 4(e^x + 2) - y = -1.
    ansln 2 sq rt. of 2, 9 + 8 sq. rt of 2

    2. Mica has six differently colored crayons. She can use one or more
    colors in her painting. What is the likelihood that she will use only her
    favorite color?
    ans: 1/63

    3. cos 15 degrees is equal to what value ?
    ans: ( sq rt of 6 + sq rt of 5 )/4

    4. A line with y-intercept 5 and positive slope is drawn such that this line
    intersects x^2 + y^2 = 9: What is the least slope of such a line?
    ans: 4/3

    5. A metal bar bent into a square is to be painted. How many distinct
    ways can one color the metal bar using four distinct colors on the edges
    using red, white, blue, and yellow.

    ans: 3

    6. In how many ways can the letters of the word MURMUR be arranged
    without letting two letters which are the same be adjacent?

    ans: 31
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  2. #2
    MHF Contributor Unknown008's Avatar
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    1. Use elimination to eliminate y. You'll then get an equation in x only.

    2. I thought of this in terms of combinations. There is only one possible favourite colour.
    There are 63 ways of choosing colours, that is:
    1 way of choosing all 6,
    6C5 ways of picking 5 of 6 colours,
    6C4 ways of picking 4 of 6 colours,
    6C3 ways of picking 3 of 6 colours,
    6C2 ways of picking 2 of 6 colours,
    6C1 ways of picking 1 of 6 colours,

    for a total of 63 ways.

    3. Use the compound formula with subtraction: \cos(A-B) = \cos A\cos B + \sin A\sin B

    4. The equation of the line is in the form: y = mx + 5

    Now, make a sketch of the curve x^2 + y^2 = 9

    This will give you an indication on about how the problem is, that is, the line is actually a tangent to the circle with equation x^2 + y^2 = 9
    Equate the two lines to get another quadratic equation.
    Use the fact that if they have equal roots, then b^2 - 4ac = 0 for ax^2 + bx + c = 0. Substitute the values and solve for a.

    5. Counting all will be the easiest method...

    6. Find the exact opposite and subtract from the total number of ways ion which the letters can be arranged.

    Post what you get!
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  3. #3
    rcs
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    Senior Member rcs's Avatar
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    i will try
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  4. #4
    Member rtblue's Avatar
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    For number 5, counting is most likely the best method, but what if you had a bigger number? What if it was a dodecagon-shaped frame with 12 different available colors?
    Counting would probably be a nasty way to go about it. Instead, we should use the formula:

    \frac{(n-1)!}{2}

    This is the formula for the total arrangements of beads(or any other object) on a keyring. This formula does apply here, even though the frame is not a key ring. When we use this we have:

    \frac{(4-1)!}{2}=3

    So we have your answer: 3 total arrangements.
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