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Question <ul><li>Mary has encountered quite the weird question on the bio test she took from her fellow SAC comrade. Maybe it’s another clue? Let’s find out! </li></ul><ul><li>Escherichia coli is a bacterium that is found in our intestines to help breakdown and digest our food. A small sample of 20 E.Coli bacteria cells was put under careful observation in a petri dish. It was found that every 2 hours, the population of bacteria tripled. </li></ul><ul><li>Correct to 4 decimal places, how many bacteria will there be after 2 weeks? </li></ul><ul><li>If the sample was made on Thursday, June 4 at 6:35:04 pm and the replication process started immediately, when would the population of the bacteria (to the nearest tenth of a second) reach 7 billion? </li></ul><ul><li>* Answer should be in the form “Weekday, Month Day Hour:Min:Sec am/pm. </li></ul><ul><li>If you put the petri dish on a piece of cm graph paper, 4 points on the graph that are touching the edge of the dish are (4, 3), (8, -1), (4, -5) and (0, -1). If each bacterial colony takes up 0.75 mm 2 of the dish’s top surface area and each bacterial colony contains 500 000 bacteria, at the time found in b), will there be an overflow? </li></ul>
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DO NOT MOVE ON UNTIL YOU HAVE ANSWERED THE QUESTION OR YOU NEED HELP!
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Things You Should Know <ul><li>Oooh, this question sounds dangerous because E. Coli is the cause of diarrhea and other disease as well. </li></ul><ul><li>This question has content from 2 units: Logarithms and Exponents and Conics. </li></ul><ul><li>Let’s start with question A. </li></ul><ul><li>Correct to 4 decimal places, how many bacteria will there be after 2 weeks? </li></ul>
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Population Count (Logs and Exp.) <ul><li>Correct to 4 decimal places, how many bacteria will there be after 2 weeks? </li></ul><ul><li>Ok, so we actually have a lot of information that’s given to us about this. We know: </li></ul><ul><li>The initial amount of bacteria (A o ) :20 </li></ul><ul><li>The growth rate or multiplication factor (m): 3 </li></ul><ul><li>The period (how long it takes for the population to be multiplied by ‘m’ once): 2 hours </li></ul><ul><li>This means we can use this formula: </li></ul><ul><li>Now we can generate the formula we need to find the population of bacteria at any time or how long it would take for a population to get to a certain size! </li></ul><ul><li>A o :20 </li></ul><ul><li>m=3 </li></ul><ul><li>p=2 </li></ul>
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Population Count (Logs and Exp.) <ul><li>Correct to 4 decimal places, how many bacteria will there be after 2 weeks? </li></ul><ul><li>Now that we have the equation, we can find out what the population will be after 2 weeks. </li></ul><ul><li>But first, we have to remember that our period is in terms of hours and we must work with common terms so we’ll convert 2 weeks into hours. </li></ul><ul><li>2 weeks(7 days/week) =14 days </li></ul><ul><li>14 days(24 hours/day) = 336 hours </li></ul><ul><li>Now that we know what 2 weeks is in hours, we can plug it into the equation! </li></ul>
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Population Count (Logs and Exp.) <ul><li>Correct to 4 decimal places, how many bacteria will there be after 2 weeks? </li></ul><ul><li>t=336 </li></ul><ul><li>A=20(3) 168 </li></ul><ul><li>A≈2.8668x10 81 bacteria </li></ul>
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Population Count (Logs and Exp.) <ul><li>Okay now for question b) </li></ul><ul><li>b) If the sample was made on Thursday, June 4 at 6:35:04 pm and the replication process started immediately, when would the population of the bacteria (to the nearest tenth of a second) reach 7 billion? </li></ul><ul><li>* Answer should be in the form “Weekday, Month Day Hour:Min:Sec am/pm. </li></ul><ul><li>Ok, so now we want to know when will the population be 7 billion (7 000 000 000). </li></ul><ul><li>First off, let’s use the equation to find how long it will take for the population to be 7 billion. </li></ul>
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Population Count (Logs and Exp.) <ul><li>A= 7 000 000 000 </li></ul>You could actually pick ‘log’ instead of ‘ln’ when you take the log of both sides. Both of these are programmed into your calculator. Don’t clear the value from your calculator. The number here is just rounded off but your calculator will store all the decimal places
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Population Count (Logs and Exp.) This value is in terms of hours meaning it will take around 35.6151 hours but we want our time in the form Day Hour:Min:Sec. So let’s break this down. The whole number is how many hours there are. There are 35 hours. There are enough hours in here for a day so let’s subtract 24 hours for the day. Now there are 11 hours left. So far we have this: 1 day 11:xx:xx.x Now we should have a decimal value less than one (0.8150803798). This value represents the fraction of an hour we have left. So let’s move down one unit of measurement. The next step down from an hour is a minute. Let’s find out how many minutes this is shall we? * All the value typed out here are rounded off. Do not round off in your calculator. Press [2 nd ][(-)] and Ans will appear on your screen. This means that the last answer you came up with will be used and you keep all the decimal places. 0.8151 hours(60 min/hour)≈48.9048 minutes Now we have 1 day 11:48:xx.x
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Population Count (Logs and Exp.) Now we have 1 day 11:48:xx.x. Now we subtract 48 so we are left with the decimal value less than one again (0.9048227887). This value represents the fraction of a minute that we have. So let’s take another step down and go to seconds! *Remember: [2 nd ][(-)] to keep the decimal places! 0.9048(60 seconds/minute)≈54.3 seconds Now we have the complete time to the nearest tenth second: 1 day 11:48:54.3 So how do we find when (date) the population will approximately be 7 billion? We add it to the date it started!
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First we convert the time into military time so that we can add it. Next we add the seconds. Now we add the minutes. Remember that there are 60 minutes in an hour so for every 60 minutes, you add 1 to the hours and start back at 0 for the minutes.
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Now we add the hours. Remember that there are 24 hours in a day so for every 24 hours, add 1 to the days and start back at 0 for the hours. Next we add the days. Great! So now we have the approximate time at which the population of the bacteria will reach 7 billion. Saturday June 6 at 6:23:58.3 am
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Population Count (Logs and Exp.) <ul><li>Okay now for question c) </li></ul><ul><li>c) If you put the petri dish on a piece of cm graph paper, 4 points on the graph that are touching the edge of the dish are (4, 3), (8, -1), (4, -5) and (0, -1). If each bacterial colony takes up 0.75 mm 2 of the dish’s top surface area and each bacterial colony contains 500 000 bacteria, at the time found in b), will there be an overflow? </li></ul><ul><li>First, let’s draw the graph. </li></ul>
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Population Count (Logs and Exp.) <ul><li>Okay now for question c) </li></ul><ul><li>c) If you put the petri dish on a piece of cm graph paper, 4 points on the graph that are touching the edge of the dish are (4, 3), (8, -1), (4, -5) and (0, -1). If each bacterial colony takes up 0.75 mm 2 of the dish’s top surface area and each bacterial colony contains 500 000 bacteria, at the time found in b), will there be an overflow? </li></ul><ul><li>Now let’s mark the points we know. </li></ul>
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Population Count (Logs and Exp.) <ul><li>Okay now for question c) </li></ul><ul><li>c) If you put the petri dish on a piece of cm graph paper, 4 points on the graph that are touching the edge of the dish are (4, 3), (8, -1), (4, -5) and (0, -1). If each bacterial colony takes up 0.75 mm 2 of the dish’s top surface area and each bacterial colony contains 500 000 bacteria, at the time found in b), will there be an overflow? </li></ul><ul><li>Now let’s connect the dots so it looks like a circle. </li></ul>
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Population Count (Logs and Exp.) <ul><li>Okay now for question c) </li></ul><ul><li>c) If you put the petri dish on a piece of cm graph paper, 4 points on the graph that are touching the edge of the dish are (4, 3), (8, -1), (4, -5) and (0, -1). If each bacterial colony takes up 0.75 mm 2 of the dish’s top surface area and each bacterial colony contains 500 000 bacteria, at the time found in b), will there be an overflow? </li></ul><ul><li>Now let’s find the center. This can be done two ways. The first is from looking at the graph. The second is by finding the midpoint between two endpoints that are opposite each other using the midpoint formula: </li></ul>
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Population Count (Logs and Exp.) <ul><li>Okay now for question c) </li></ul><ul><li>c) If you put the petri dish on a piece of cm graph paper, 4 points on the graph that are touching the edge of the dish are (4, 3), (8, -1), (4, -5) and (0, -1). If each bacterial colony takes up 0.75 mm 2 of the dish’s top surface area and each bacterial colony contains 500 000 bacteria, at the time found in b), will there be an overflow? </li></ul><ul><li>Next we find the radius. The radius would be the distance from the center out to one of the endpoints. The radius of this circle is 4 cm. </li></ul>
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Population Count (Logs and Exp.) <ul><li>Okay now for question c) </li></ul><ul><li>c) If you put the petri dish on a piece of cm graph paper, 4 points on the graph that are touching the edge of the dish are (4, 3), (8, -1), (4, -5) and (0, -1). If each bacterial colony takes up 0.75 mm 2 of the dish’s top surface area and each bacterial colony contains 500 000 bacteria, at the time found in b), will there be an overflow? </li></ul><ul><li>Bonus question! From here, what would the equation for the circle be? </li></ul>
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Population Count (Logs and Exp.) <ul><li>Okay now for question c) </li></ul><ul><li>c) If you put the petri dish on a piece of cm graph paper, 4 points on the graph that are touching the edge of the dish are (4, 3), (8, -1), (4, -5) and (0, -1). If each bacterial colony takes up 0.75 mm 2 of the dish’s top surface area and each bacterial colony contains 500 000 bacteria, at the time found in b), will there be an overflow? </li></ul><ul><li>Answer: </li></ul><ul><li>The form for the equation of a circle is: </li></ul><ul><li>(x-h) 2 +(y-k) 2 =r 2 </li></ul><ul><li>where </li></ul><ul><li>(h,k) is the center </li></ul><ul><li>r is the radius </li></ul><ul><li>So the equation for this circle is: </li></ul><ul><li>(x-4) 2 +(y+1) 2 =4 2 </li></ul>
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Population Count (Logs and Exp.) <ul><li>Okay now for question c) </li></ul><ul><li>c) If you put the petri dish on a piece of cm graph paper, 4 points on the graph that are touching the edge of the dish are (4, 3), (8, -1), (4, -5) and (0, -1). If each bacterial colony takes up 0.75 mm 2 of the dish’s top surface area and each bacterial colony contains 500 000 bacteria, at the time found in b), will there be an overflow? </li></ul><ul><li>Ok back to business. Now we have to find the area of this circle. To find the area of a circle, it’s: </li></ul><ul><li>A= π r 2 </li></ul><ul><li>A= π r 2 </li></ul><ul><li>A= π 4 2 </li></ul><ul><li>A= π (16) </li></ul><ul><li>A≈50.2655 cm 2 </li></ul>
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Population Count (Logs and Exp.) <ul><li>Okay now for question c) </li></ul><ul><li>c) If you put the petri dish on a piece of cm graph paper, 4 points on the graph that are touching the edge of the dish are (4, 3), (8, -1), (4, -5) and (0, -1). If each bacterial colony takes up 0.75 mm 2 of the dish’s top surface area and each bacterial colony contains 500 000 bacteria, at the time found in b), will there be an overflow? </li></ul><ul><li>A≈50.2655 cm 2 </li></ul><ul><li>Let’s convert this measurement to mm 2 . (The number typed is rounded. Use actual value on calculator.) </li></ul><ul><li>50.2655cm 2 (100mm 2 /cm 2 )≈5026.5482 mm 2 </li></ul><ul><li>Ok, so now we know how the area of the circle. </li></ul><ul><li>Lets find the area our 7 billion bacteria take up. </li></ul>
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Population Count (Logs and Exp.) <ul><li>Okay now for question c) </li></ul><ul><li>c) If you put the petri dish on a piece of cm graph paper, 4 points on the graph that are touching the edge of the dish are (4, 3), (8, -1), (4, -5) and (0, -1). If each bacterial colony takes up 0.75 mm 2 of the dish’s top surface area and each bacterial colony contains 500 000 bacteria, at the time found in b), will there be an overflow? </li></ul><ul><li>First we find the number of colonies there are. </li></ul><ul><li>7 000 000 000 bacteria(1 colony/500 000 bacteria) = 14 000 colonies </li></ul><ul><li>Now we find out how much area the colonies cover. </li></ul><ul><li>14 000 colonies (0.75 mm 2 /colony) = 10 500 mm 2 </li></ul><ul><li>So the colonies we have will take up 10 500 mm 2 . Wait. We only have ≈5026.5482 mm 2 of space! That means…… </li></ul>
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OH NO! THERE WILL BE A BACTERIAL OVERFLOW!!!!!!!
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