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Math Help - Argh. I hate word problems. Help?

  1. #1
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    Argh. I hate word problems. Help?

    Yes, I know I should do my own work and such, but I really don't get some of these word problems; I did all of them but three.
    I'm really sorry for bothering the forum, but I didn't know where else to get help from, my Pre-Calculus teacher basically just assigns us a chapter to read and then expects us to be able to do the assignments and tests. Well, that may do it for some people, but not for me.

    If you have read this far, thank you so much!

    These are the word problems I have left to do:

    Sales on baseball caps average 31000 per game when they are priced at $2.50 each. For each .50 increase, sales drop by 2000; what price would bring in the maximum revenue?

    A rectangular piece of cardboard is measured to be 25 inches by 39 inches, it is then made into a box by cutting equally sized squares from each corner and folding it up. What can the maximum volume be?

    I'm currently figuring this one out, but any help will be appreciated:
    The perimeter of a rectangle is 36 inches. With the width becoming doubled and the length being increased by 21 inches, the new perimeter is 92 inches. What are the measurements of the length and width?

    Thank you again so much!
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  2. #2
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    Quote Originally Posted by LordTacodip View Post
    Sales on baseball caps average 31000 per game when they are priced at $2.50 each. For each .50 increase, sales drop by 2000; what price would bring in the maximum revenue?
    1) Calm down and quit blaming the world.
    2) Forget you ever heard the words "word problem". For some reason, these two words, juxtaposed, put great fear int he hearts of many. You seem to be one of them.
    3) Learn to approach the problem deliberately and sequentially.
    4) The following shoudl not be scary. Just drag through the problem one hint at a time. Think about it at every step. Write clear and concise defintions along the way.

    "Sales on baseball caps average 31000 per game when they are priced at $2.50 each."

    Pretty obviously

    C = Number of Caps Sold
    2.50 * C = 31,000
    C = 12,400

    "For each .50 increase, sales drop by 2000"

    Great:

    (2.50 + 0.50)*(12,400 - 2000) = (3.00)(10,400) = 31,200
    It went up!

    (2.50 + 2*0.50)*(12,400 - 2*2000) = (3.50)(8,400) = 29,400
    Whoops! It went down.

    Or more generally, we'll need a difinition of the number of increases:

    d = # of 50 increases.

    (2.50 + d*0.50)*(12,400 - d*2000) = ??

    A little algebra produces

    31,000 - 5,000*d + 6,200*d - 1000*d*d = f(d)

    or

    f(d) = -1000d^{2} + 1200d + 31000

    You should recognize this as a downward-opening parabola and you should be well-trained to find it's maximum.

    -1200/(-2000) = 0.6

    Then, f(0.6) should be the maximum. Is it?

    Since d is defined as the number of 50 price increases, we have a price of 2.50 + 0.60*0.50 = 2.80.
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  3. #3
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    First of all, thank you so much for replying to my original post! I see where I went wrong, I was forming my function wrong, thus messing up the upside down parabola when I would graph it with my calculator.
    I wasn't blaming anyone by the way, I'm sorry if it sounded like that. I just mentioned that my Pre-Calculus teacher's teaching methods don't exactly line up on how I learn math, that's all. He's a real cool person, but I'm not exactly competent at teaching myself the math (we get a lot of self-study time.)

    I just find it hard to translate a word problem into a function, that's what I have problems. I'm sorry for asking so much, but could someone walk me through with the cardboard problem?
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  4. #4
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    It would be most beneficial to see you walk through it. Let's see what you get.

    Hint: Why do you care about the perimeter? Focus on the volume.
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  5. #5
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    In regards to the rectangular cardboard box, cutting a square of side length x from each corner decreases the length of each side of the rectangle by 2x (because each side has 2 corners from each of which a square of side length x is being removed).

    Once the squares are cut from each corner, the rectangle measures 25 - 2x by 39 - 2x.

    Once folded, the volume is given by length times width times height. Find the maximum of the volume. I'll let you try to solve it from here.



    In regards to finding the dimensions of the rectangle, the perimeter is given by the sum of twice the length and twice the width.

    P = 2l + 2w

    The perimeter is given as 36. Therefore, 36 = 2l + 2w.

    If the length is increased by 21 and the width is doubled, then the new perimeter is given by P = 2(l + 21) + 2(2w) = 2l + 42 + 4w = 2l + 4w + 42.

    The new perimeter is given as 92. Therefore, 92 = 2l + 4w + 42.

    Solve the system of equations to find the length and width.

    36 = 2l + 2w

    92 = 2l + 4w + 42

    I'll let you try to solve it from here.
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