# HW 7/9

1. Snail Invasion: https://www.illustrativemathematics.org/illustrations/638

a. What was the initial number of snails?  Do you have other known values of an output at a certain time of years?  Create an equation to solve for the rate of growth, and create an equation of the form y=a•c^x where a and c are given, or y=a•e^rx where a and r are given.

b. What is the initial amount of the population?  What is it when it doubles?  Solve for the time it takes to double using the equation you created in part a.

c. The cost being proportional to the number of snails means that from \$1,000,000 for 18,000 snails, you can deduce the “per snail cost,” and use that to calculate costs for different quantities of snails.  What was the number of snails the year before there were 18000?  What would it be the following year?

2. Illegal Fish: https://www.illustrativemathematics.org/illustrations/579

a. The amount of fisherman released into the lake would be considered the initial value, found at week 0.

b. You know that there are 33 fish at a time of 8 weeks.  You also know the initial value from part a.  Plug that information into the equation and find the rate of growth (b).

c.  Use the structure of the equation to determine the rate of growth, but just be sure to phrase it in terms of percent.  If you are having trouble, consider some specific cases, for example from week 2 to week 3.

3. US Population 1982-1988: https://www.illustrativemathematics.org/illustrations/353

a. Think about what the graph of a linear equation looks like: it keeps going up, never stopping.  Is that what populations do?  Or do they level off at some point?  What kind of function would describe values that level off as time goes on?  Maybe there is a specific domain of time that a linear model would be good for.

b. What is another word for slope?  What does that mean in the context of this problem?

c. Before you can use Mike’s model, you have to know what it is.  Remember: it is a linear model: y=mx+b.  You already know the slope.  Let x represent the number of years after 1982, and y represent the population of that year.  Hint: 1982 is 0 years after 1982, and you know the population that year!

a. Fill out the table.  1800mm is height you drop the ball from: no bounces have a occurred.  After one bounce, the ball reaches what height?  What is the ratio from the initial height to the height after one bounce?  Remember, to get each consecutive height, the ratio stays the same!

b.) What do you multiply the first output by to get the second output?  That is your multiplicative rate of change.  You should also know the initial value.

c.) You want to consider the scenario where your output is less than 100mm.  Use the equation you developed in part b. Note: you can’t actually have fractional bounces, so consider that when you interpret your answer.

5. Bacteria Populations: https://www.illustrativemathematics.org/illustrations/370

a. Use the structure of the expressions to determine the initial values and rate of growth of each function.  To interpret this, consider this scenario: If Tara swims faster than me, she might give me a head start in a swimming race.  So my initial value is higher.  But her swimming rate is faster, so she will likely catch up and even pass me.  Now consider what would happen if I gave Tara a head start.  She starts ahead of me, and she already swims faster than me! Will I ever catch up?  Will our paths ever intersect?

b. What does it mean for two functions to have the same value at the same time?  They must be equal!  Set the two expressions equal to each other.  Be careful when you solve: Consider this example:

ln(m•n)= ln(m)+ln(n). BUT: ln(m•n^2) is NOT 2•ln(m•n).  I can’t bring the exponent of one variable down in front of the whole logarithm. ln(m•n^2)=ln(m)+ln(n^2), which can then be simplified as ln(m)+2•ln(n).  Be careful with your exponents and what rules you apply to them.

6. Newton’s Law of Cooling: https://www.illustrativemathematics.org/illustrations/382

Look in your notes for our coffee example to help interpret this problem!

7. Interesting Interest Rates: https://www.illustrativemathematics.org/illustrations/302

a.) Be careful about simple interest vs. compound interest.  How much interest does the simple interest bank earn every year?  Remember: simple interest doesn’t change from year to year.  Even though there is more money in your bank due to interest, you earn the same interest every year.  For compound interest, use A=P(1+r/n)^nt.

b.) C(y), the function for City Bank, is going to be a linear model! Think about your initial amount, and how much gets added per year (y), and create an equation representing this.  Also note for N(m) that the time m, is being measured in months, NOT years! The equation you probably used for part a will have to be adjusted slightly.  What is the monthly interest rate?  m then represents how many times that monthly interest rate is compounded.

c. Just create your table of values year by year!  Be careful about which functions you use though: If you use your monthly function from part b, you will have to adjust accordingly.

8. Exponential Growth vs. Linear Growth 1: https://www.illustrativemathematics.org/illustrations/366

a. What is the difference between doubling each consecutive output, vs. adding two?  What different categories of functions do those scenarios represent?

b. Create an equation representing each method.  Find out where they intersect, and then try a value after that solution.

9. Lake Algae: https://www.illustrativemathematics.org/illustrations/533

a. Hint: If my money doubles everyday, and I have \$100 today, how much did I have yesterday?

b. Determine exactly what fraction of the lake is actually covered on June 16 to give a good answer.

c. To say how well this solves the problem, determine how long it will take for the lake to get covered again.

d. What kind of function is this going to be? Linear?  Quadratic? Exponential?  You are going to need to find the initial value.  What do you know?  Rate? Do you have a known amount on a known day number?

10. Algae Blooms: https://www.illustrativemathematics.org/illustrations/570

a. When a cell divides, what is the new quantity?  How would you describe the rate of change?  What is the initial condition given?

b. Use the equation developed in part a.  What constitutes a “bloom” according to the description in the problem?

c.  If one cell divides, and then those cells divide, how many cells are there?  What is that rate of change?  For this problem, assume the same initial condition as part a.

11. Carbon 14 Dating: https://www.illustrativemathematics.org/illustrations/369

a. Consider your initial amount, you can leave it as a variable or use a specific number.  What ever you choose, the output for the half-life of the atom is going to be half of the initial value.  Set up this equation and solve for c.

b. Currently having 1 microgram is the output of your function.  Solve for time.

12. Carbon 14 Dating in Practice 1: https://www.illustrativemathematics.org/illustrations/758

a. The plant currently has 1 microgram of Carbon.  That is not the initial value. That is the value after 5000 years.  Solve for the initial amount using the given information.

b. Similar to a.

c. How many years does it take for the initial value to turn into half of the initial value?  You can choose a specific value for the initial value, and base your output off of that, and then solve for the number of years that satisfies the equation.

a. Note: The interest is only paid at the end of each year, it is not split up over any intervals, but it is still compounded.

b. Note: In this problemn does NOT stand for the number times the interest is split over.  The interest is not split up!  n stands for the number of times the interest is calculated.  How often is interest applied to the account?  Where does n belong in this equation?

c. You’ve got this…

d. The key here is that the problem is using r to represent a percentage, whereas we used r as a decimal in class.  So the r/100 is to convert r from a percentage to a decimal.  The divided by 100 is not splitting the interest rate into 100 intervals!  100 is NOT n!  Consider: What is your initial amount, what is the output, what is your rate, what is your time.  Leave r as r, and as you manipulate the equation, I think you will see what the question is getting at.

14. A Valuable Quarter: https://www.illustrativemathematics.org/illustrations/1421

a. You know how to do these problems!

b. Compare the current value of the coin to the values achieved through its growth in the bank at the rates considered in problem a.  What’s the range of rates that the coin’s value falls between?

c. What was the value of the quarter when it was originally created?  (This is not a trick question!)  Use the value of the coin today and figure out the yearly compounded interest rate that got the coin from its initial value to its current value over the course of how many years it’s been.

For all of these problems, remember the general form for exponential equations:

y=a•c^x, where a is the initial value, c is the multiplicative rate of change, and x is the number of times the rate occurs (days, minutes, hours, weeks, years…)

y=a•e^rt is for continuos growth or decay.  For decay (where values are decreasing), r is going to be negative.  t stands for the number of cycles the rate is applied.  So if your rate is yearly, t is measured in years.  a once again is the initial value, when t=0.

For A=P(1+r/n)^nt, n is the number of times that the interest rate is divided into over the course of a year.  If the interest rate is only applied yearly, don’t worry about n, it equals 1. In this formula, r is assumed to be a decimal, not a percent.

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