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
According to a company's website, the top 15% of the candidates who take the entrance test will be called for an interview. You
galben [10]

Answer:

The minimum score required for an interview is 77.252

Step-by-step explanation:

We solve this using z score formula

z = (x-μ)/σ, where x is the raw score, μ is the population mean, and σ is the population standard deviation.

Top 15% of the candidates is a ranking that is equivalent to = 100 - 15% = 85th percentile.

The z score of 85th percentile = 1.036

Mean = 70

Standard Deviation = 7

Minimum score = raw score = ???

Hence:

1.036 = x - 70/7

Cross Multiply

1.036 × 7 = x - 70

7.252 = x - 70

x = 70 + 7.252

x = 77.252

The minimum score required for an interview is 77.252

6 0
3 years ago
Find the annual interest rate. I=$16I=$16, P=$200P=$200, t=2t=2 years
BigorU [14]
R=i/pt
R=16/200*2=0.04*100=4%
3 0
3 years ago
If a car travels at a speed of 25 mi/h for t hours, then travels 45 mi/h for m hours. What does the expression 25t + 45m represe
dlinn [17]
The expression of 25t +45m shows the total number of miles driven at both speeds. In order to solve this equation you would need to know the total number of hours driven in order to multiply that by the appropriate speed in terms of miles per hour.
6 0
3 years ago
The money raised By $15 candy is approximately What percent of The total money raised from The candy sale
Monica [59]
Can you explain it a bit more?
7 0
3 years ago
Read 2 more answers
Is anyone knows the answer? Thanks. What line has a slope of 0 and goes through (5, -8)
larisa [96]
Slope = 0
⇒ It is a horizontal like that cuts through the y-axis

At point (5, -8), y = -8
The equation of the line is y = -8

-----------------------------------------------
Answer: y = -8
-----------------------------------------------
8 0
3 years ago
Other questions:
  • Purslane bought 350 rolls of toilet paper for the whole year. How many rolls of toilet paper per month is that?
    7·2 answers
  • The geometric mean between 1/2 and 2/5 <br> Thanks!
    8·1 answer
  • In how many distinct ways an the letters of the word Atlanta be arranged
    14·1 answer
  • Which equation shows P=2l + 2w when solved for w?
    15·2 answers
  • Determine the intercepts of the line.
    13·2 answers
  • If 7x+2a = 3x+5a, then x is equivalent to
    12·1 answer
  • (01.02 LC)Simplify 9 − (−4).
    9·2 answers
  • The cost d of a hotdog is $1 more than half the cost of a sandwich s.
    12·1 answer
  • Carmen has bought 30 pounds of dog food. She feeds her dog 2/3 pounds for each meal. For how many meals will the food last?
    13·2 answers
  • 4. A diagonal divides a square patch of garden that is
    15·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!