Solved on Dec 04, 2023

Find an exponential model $P=a\cdot b^t$ for a population of 16,000 organisms growing by $6.9\%$ annually.

STEP 1

Assumptions
1. The initial population is 16,000 organisms.
2. The growth rate is 6.9% per year.
3. The population growth can be modeled exponentially.
4. The exponential model will be in the form $P = a \cdot b^{t}$ , where: - $P$ represents the population after $t$ years, - $a$ represents the initial population, - $b$ represents the growth factor, - $t$ represents the number of years.

STEP 2

Identify the initial population, which will be the value of $a$ in the exponential model.
$a = 16,000$

STEP 3

Convert the growth rate from a percentage to a decimal to find the growth factor $b$ .
$Growth\, rate = 6.9\% = 0.069$

STEP 4

Calculate the growth factor $b$ by adding 1 to the decimal growth rate. This is because the growth factor represents the total population after 1 year, which includes both the original population and the growth.
$b = 1 + Growth\, rate$

STEP 5

Plug in the growth rate in decimal form to find the growth factor $b$ .
$b = 1 + 0.069$

STEP 6

Calculate the growth factor $b$ .
$b = 1 + 0.069 = 1.069$

STEP 7

Write the exponential model using the values of $a$ and $b$ that we have found.
$P = a \cdot b^{t}$

STEP 8

Substitute the values of $a$ and $b$ into the model.
$P = 16,000 \cdot 1.069^{t}$
The exponential model for the population growth is: $P = 16,000 \cdot 1.069^{t}$

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Find an exponential model $P=a\cdot b^t$ for a population of 16,000 organisms growing by $6.9\%$ annually.

STEP 1

STEP 2

Identify the initial population, which will be the value of $a$ in the exponential model.
$a = 16,000$

STEP 3

Convert the growth rate from a percentage to a decimal to find the growth factor $b$ .
$Growth\, rate = 6.9\% = 0.069$

STEP 4

Calculate the growth factor $b$ by adding 1 to the decimal growth rate. This is because the growth factor represents the total population after 1 year, which includes both the original population and the growth.
$b = 1 + Growth\, rate$

STEP 5

Plug in the growth rate in decimal form to find the growth factor $b$ .
$b = 1 + 0.069$

STEP 6

Calculate the growth factor $b$ .
$b = 1 + 0.069 = 1.069$

STEP 7

Write the exponential model using the values of $a$ and $b$ that we have found.
$P = a \cdot b^{t}$

STEP 8

Substitute the values of $a$ and $b$ into the model.
$P = 16,000 \cdot 1.069^{t}$
The exponential model for the population growth is: $P = 16,000 \cdot 1.069^{t}$

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Find an exponential model P=a⋅btP=a\cdot b^tP=a⋅bt for a population of 16,000 organisms growing by 6.9%6.9\%6.9% annually.

STEP 1

STEP 2

Identify the initial population, which will be the value of a a a in the exponential model.a=16,000 a = 16,000 a=16,000

STEP 3

Convert the growth rate from a percentage to a decimal to find the growth factor b b b.Growth rate=6.9%=0.069 Growth\, rate = 6.9\% = 0.069 Growthrate=6.9%=0.069

STEP 4

Calculate the growth factor b b b by adding 1 to the decimal growth rate. This is because the growth factor represents the total population after 1 year, which includes both the original population and the growth.b=1+Growth rate b = 1 + Growth\, rate b=1+Growthrate

STEP 5

Plug in the growth rate in decimal form to find the growth factor b b b.b=1+0.069 b = 1 + 0.069 b=1+0.069

STEP 6

Calculate the growth factor b b b.b=1+0.069=1.069 b = 1 + 0.069 = 1.069 b=1+0.069=1.069

STEP 7

Write the exponential model using the values of a a a and b b b that we have found.P=a⋅bt P = a \cdot b^{t} P=a⋅bt

STEP 8

Substitute the values of a a a and b b b into the model.P=16,000⋅1.069t P = 16,000 \cdot 1.069^{t} P=16,000⋅1.069tThe exponential model for the population growth is: P=16,000⋅1.069t P = 16,000 \cdot 1.069^{t} P=16,000⋅1.069t

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Find an exponential model $P=a\cdot b^t$ for a population of 16,000 organisms growing by $6.9\%$ annually.

Identify the initial population, which will be the value of $a$ in the exponential model.
$a = 16,000$

Convert the growth rate from a percentage to a decimal to find the growth factor $b$ .
$Growth\, rate = 6.9\% = 0.069$

Calculate the growth factor $b$ by adding 1 to the decimal growth rate. This is because the growth factor represents the total population after 1 year, which includes both the original population and the growth.
$b = 1 + Growth\, rate$

Plug in the growth rate in decimal form to find the growth factor $b$ .
$b = 1 + 0.069$

Calculate the growth factor $b$ .
$b = 1 + 0.069 = 1.069$

Write the exponential model using the values of $a$ and $b$ that we have found.
$P = a \cdot b^{t}$

Substitute the values of $a$ and $b$ into the model.
$P = 16,000 \cdot 1.069^{t}$
The exponential model for the population growth is: $P = 16,000 \cdot 1.069^{t}$