# Thread: 2 problem solving questions

1. ## 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? 2. Originally Posted by johntuan 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 $550 and have 96 apartments. So the total income is$\displaystyle 96 \cdot \$550 = \$ 52800$which is$1800 short of expectations.

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

Since we need an income of $54600 we will need:$\displaystyle 54600 = 52800 + 750n - 75n^2$or$\displaystyle 75n^2 - 750n + 1800 = 0\displaystyle n^2 - 10n + 24 = 0\displaystyle (n - 4)(n - 6) = 0$Thus n = 4 or 6. So we are using either 4 steps of$25 or 6 steps of $25. Which value should you choose? I'll leave that one open to the more business minded people than me. -Dan 3. Originally Posted by johntuan 2. A health clinic has a solution of bleach to sterilize petri dishes in which culture are grown. The sterilization tank contains 100 gal of a solution of 2% ordinary household bleach mixed with pure distilled water. Now research indicated that concentration of bleach should be %5 for complete sterilisation. How much of the solution should be drain and replaced with bleach content to the recommended level? In every 100 gal of 2% solution there are 2 gal of pure bleach. We wish to remove some number of gallons x of solution and replace it with x gal of pure bleach, such that the solution concentration is 5%. So we have 100 - x gal of 2% solution + x gal bleach which gives us the 5% solution. By volume this will give 100 gal of 5% solution, so we expect there to be a total of 5 gal of pure bleach. In 100 - x gal of 2% solution there are 0.02(100 - x) gal of pure bleach. So:$\displaystyle 0.02(100 - x) + x = 5\displaystyle 2 - 0.02x + x = 5\displaystyle 0.98x = 3\displaystyle 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 4. Originally Posted by topsquark 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? 5. Originally Posted by johntuan 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 6. 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.