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Math Help - 2 problem solving questions

  1. #1
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    2 problem solving questions

    1. A real-state firm owns the Parklane Garden Apartments, which consists of 96 apartments. At $550 per month, every apartment can be rented. However, for each $25 per month increase, there will be three vacancies with no possibility of filling them. The firm wants to recieve $54,600 per month from rents. What rent should be charged for each apartment?

    2. A health clinic has a solution of bleach to sterilize petri dishes in which culture are grown. The sterilization tank contains 100 gal of a solution of 2% ordinary household bleach mixed with pure distilled water. Now research indicated that concentration of bleach should be %5 for complete sterilisation. How much of the solution should be drain and replaced with bleach content to the recommended level?
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  2. #2
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    Quote Originally Posted by johntuan View Post
    1. A real-state firm owns the Parklane Garden Apartments, which consists of 96 apartments. At $550 per month, every apartment can be rented. However, for each $25 per month increase, there will be three vacancies with no possibility of filling them. The firm wants to recieve $54,600 per month from rents. What rent should be charged for each apartment?
    Not getting too fancy about it, but you start with a base rent of $550 and have 96 apartments. So the total income is 96 \cdot \$ 550 = \$ 52800 which is $1800 short of expectations.

    So by increasing the rent by $25 we lose 3 apartments. So the income may be calculated by:
    I = (96 - 3n)( \$ 550 + \$ 25 \cdot n)
    where n is the number of $25 steps the rent has increased by. Numerically we have:
    I = (96 - 3n)(550 + 25n) = 52800 + 750n - 75n^2

    Since we need an income of $54600 we will need:
    54600 = 52800 + 750n - 75n^2
    or
    75n^2 - 750n + 1800 = 0

    n^2 - 10n + 24 = 0

    (n - 4)(n - 6) = 0

    Thus n = 4 or 6.

    So we are using either 4 steps of $25 or 6 steps of $25. Which value should you choose? I'll leave that one open to the more business minded people than me.

    -Dan
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by johntuan View Post
    2. A health clinic has a solution of bleach to sterilize petri dishes in which culture are grown. The sterilization tank contains 100 gal of a solution of 2% ordinary household bleach mixed with pure distilled water. Now research indicated that concentration of bleach should be %5 for complete sterilisation. How much of the solution should be drain and replaced with bleach content to the recommended level?
    In every 100 gal of 2% solution there are 2 gal of pure bleach. We wish to remove some number of gallons x of solution and replace it with x gal of pure bleach, such that the solution concentration is 5%.

    So we have 100 - x gal of 2% solution + x gal bleach which gives us the 5% solution. By volume this will give 100 gal of 5% solution, so we expect there to be a total of 5 gal of pure bleach.

    In 100 - x gal of 2% solution there are 0.02(100 - x) gal of pure bleach. So:
    0.02(100 - x) + x = 5

    2 - 0.02x + x = 5

    0.98x = 3

    x = \frac{3}{0.98} \approx 3.0612 gal

    So we need to remove 3.0612 gal of the 2% solution and replace it with 3.0612 gal of pure bleach.

    -Dan
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    Quote Originally Posted by topsquark View Post

    So we are using either 4 steps of $25 or 6 steps of $25. Which value should you choose? I'll leave that one open to the more business minded people than me.

    -Dan
    Ive gotten everything that you have said but what do i do now? Do I take 4*25 or 6*25 then add it to the rent and thats my answer?
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  5. #5
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by johntuan View Post
    Ive gotten everything that you have said but what do i do now? Do I take 4*25 or 6*25 then add it to the rent and thats my answer?
    Yup!

    So the rent will either be $650 or $700 depending on how many empty apartments the place wants.

    -Dan
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  6. #6
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    thanks for your help. Imo it would make more business sense to charge $650 per month since you are housing more people.Thus, you can make more profits if you increase the rent. I could be wrong though.
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