1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Ne4ueva [31]
2 years ago
14

The population of a pigeons in a city is 100 and is growing exponentially at 20% per year. Write a function to represent the pop

ulation of pigeons after
t
t years, where the monthly rate of change can be found from a constant in the function. Round all coefficients in the function to four decimal places. Also, determine the percentage rate of change per month, to the nearest hundredth of a percent
Mathematics
2 answers:
NikAS [45]2 years ago
5 0

Answer:

  • p(t) = 100×1.0153^(12t)
  • 1.53% per month

Step-by-step explanation:

In general, the function will be written ...

  p(t) = (initial value)×(growth factor)^t

where t is in units comparable to those applicable to the growth factor. The growth factor is found from ...

  growth factor = 1 + growth rate

Here, the growth rate is given as 20% per year, so the growth factor per year is ...

  1 +20% = 1.20

The initial value is given as 100, so we can write the exponential function as ...

  p(t) = 100×1.20^t

__

The time period units for t are supposed to be years, but we want to find the growth rate for a month. We can do that by recognizing there are 12 months in a year. In the above equation, we can use (1/12)(12t) in place of t, then figure the growth factor (and growth rate) per month.

  p(t) = 100×(1.20^(1/12))^( 12t)

  p(t) = 100×1.0153^(12t) . . . . population exponential function

This shows the monthly growth factor is 1.0153, so the monthly rate of change (growth rate) is ...

  1.0153 -1 = 0.0153 = 1.53% . . . . monthly rate of change

Andrews [41]2 years ago
3 0

Answers:

  • Function is P(t) = 100(1.0153)^{12t}\\\\
  • Monthly growth rate is roughly <u>1.53%</u>

=====================================================

Explanation:

The population starts at 100 pigeons. After a year, the population goes up by 20%, meaning the population goes up by 20. It lands on 120 after a full year (100+20 = 120).

Dividing the "before" and "after" population amounts gets us 120/100 = 1.20

If we didn't worry about the monthly rate, and only cared about the annual rate of growth, then the population function would be

P(t) = 100(1.20)^t

where t is the number of years and P(t) is the population for that year number. Notice how the 1.20 is the base of the exponential to quickly tell the reader the annual rate of growth. We read off the .20 portion and that converts to 20% after moving the decimal 2 spots to the right.

If we were to plug t = 0 into that function, then we'd get P(0) = 100. Inputting t = 1 leads to P(1) = 120. This helps confirm we have the correct annual growth rate function.

---------------------

Unfortunately, your teacher doesn't want the annual growth rate function. Instead, they want you to write a function where the reader can easily pick out the monthly rate of change without having to do any math. This is due to the instructions that "the monthly rate of change can be found from a constant in the function". Let m be this constant.

Since t is the number of years, this means t = 1/12 represents 1/12 of a year, aka 1 month. If you were to compute P(1/12), then you should get roughly 101.530947049973

Divide this amount over the initial population to find m.

m = 101.530947049973/100 = 1.01530947049973

So we get m = 1.0153 after rounding to four decimal places.

The number m is in the form 1+x. Solving 1+x = m will lead to x = 0.0153 to indicate a monthly rate of change is roughly 1.53% since we move the decimal point 2 spots to the right to go from 0.0153 to 1.53%

The pigeon population is increasing by roughly 1.53% each month.

This is why computing this m value leads to the reader able to spot the monthly rate of change without having to resort to a calculator. All they have to do is read the digits to the right of the decimal point and convert that to percentage form.

---------------------

An alternative way to compute this value is to use this formula

m = (1+r)^(1/12)

We'll plug in the annual rate r = 0.20 to get:

m = (1+r)^(1/12)

m = (1+0.20)^(1/12)

m = 1.01530947049973

m = 1.0153

We get the same approximate value as earlier.

---------------------

Whichever method you use, you should get roughly m = 1.0153

We're given that A = 100 is the initial population

So we'll then be able to form this function

P(t) = A*(m)^{12t}\\\\P(t) = 100(1.0153)^{12t}\\\\

This represents the monthly growth of the population where t is in years. Note that after something like t = 3 years, we have 12t = 12*3 = 36 months pass by. So that's why the 12t is in the exponent rather than simply t.

Once again, for comparison we have...

  • P(t) = 100(1.20)^{t}\\\\ for annual growth
  • P(t) = 100(1.0153)^{12t}\\\\ for monthly growth

Each function has t represent the number of years. The difference is the base of each exponential to tell us the respective annual or monthly growth rate. Also, the exponent up top must be adjusted depending on which you go for.

---------------------

You may be wondering: "what if the exponent in the monthly growth rate function wasn't 12t?". Let's find out. We'll assume the monthly growth rate function is P(t) = 100(1.0153)^{t}\\\\ and show that an error happens.

If we plugged in something like t = 1, we should get P(t) = 120 like mentioned earlier.

P(t) = 100(1.0153)^{t}\\\\P(1) = 100(1.0153)^{1}\\\\P(1) = 100(1.0153)\\\\P(1) = 101.53\\\\

That value of 101.53 is the approximate population number after 1 month, rather than 1 year. Making the exponent 12t will fix the issue.

You might be interested in
A newsgroup is interested in constructing a 90% confidence interval for the proportion of all Americans who are in favor of a ne
Leto [7]

Answer:

The 90% confidence interval for the proportion of all Americans who are in favor of a new Green initiative is (0.6247, 0.6923).

Step-by-step explanation:

In a sample with a number n of people surveyed with a probability of a success of \pi, and a confidence level of 1-\alpha, we have the following confidence interval of proportions.

\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}

In which

z is the z-score that has a p-value of 1 - \frac{\alpha}{2}.

Of the 533 randomly selected Americans surveyed, 351 were in favor of the initiative.

This means that n = 533, \pi = \frac{351}{533} = 0.6585

90% confidence level

So \alpha = 0.1, z is the value of Z that has a p-value of 1 - \frac{0.1}{2} = 0.95, so Z = 1.645.

The lower limit of this interval is:

\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.6585 - 1.645\sqrt{\frac{0.6585*0.3415}{533}} = 0.6247

The upper limit of this interval is:

\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.6585 + 1.645\sqrt{\frac{0.6585*0.3415}{533}} = 0.6923

The 90% confidence interval for the proportion of all Americans who are in favor of a new Green initiative is (0.6247, 0.6923).

6 0
3 years ago
HELP PLEASE!!!! WILL GIVE 100 POINTS
Artist 52 [7]

Answer:

w to r to a

Step-by-step explanation:

e to c to p

8 0
3 years ago
Read 2 more answers
Please help will give brainliest
notka56 [123]

Answer:

b

the points make a sad face on the graph  and if you do the vertical line test you can see that its a function

3 0
3 years ago
A one pound bag of skittles contains 300 pieces of which 28% are red. How many skittles were red?
lesantik [10]

Answer:84

Step-by-step explanation:

3 0
3 years ago
-1/2m=-9 what does m=
lilavasa [31]

Answer:

m = 18

Step-by-step explanation:

Step 1: Write out equation

-1/2m = -9

Step 2: Multiply both sides by -2

m = 18

5 0
3 years ago
Other questions:
  • What is 0.4 as a fraction in lowest term
    9·2 answers
  • State the domain for the set of points.<br> {(8, 8), (5, 7), (-4, 8), (4, 8)}
    14·1 answer
  • 17/64 + 47/64 + 7/8 =<br>B.<br>1/16<br>134<br>151/64<br>1%<br>D.​
    10·1 answer
  • PLEASE HELP ME! What is the equation of a line that passes through the point (6, 1) and is perpendicular to the line whose equat
    9·1 answer
  • Dexter's dad runs a toy store. His current yearly sales report shows that he sold 4,694 toys in the last three months. If the st
    8·2 answers
  • F(x) = x^2– 2x + 3; f(x) = –2x + 12
    6·1 answer
  • Julio places gray chairs and blue chairs in each row the ratio of gray to blue chairs in each row 7-3 how many gray chairs and b
    12·1 answer
  • I need these answers!!! will give branliest
    11·1 answer
  • 42. The price of a baseball cap is $9.00 plus $0.78<br> tax. What is the price of 4 caps?
    6·2 answers
  • The slope of the line is 4. which of the following is the point-slope form of the line
    5·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!