Type the correct answer in each box. Use numerals instead of words. Sabrina is researching the growth of a population of horses on a ranch. She models the population of horses using the function below, where $n$ is the number of years after she begins the research and $b$ is an unknown base. [tex]w(n)=15 \cdot b^n[/tex] Based on the model, the initial number of horses is [ ]. If [tex]b=1.35[/tex], the annual percent growth rate of the number of horses would be [ ]%.
Answer (2)
The initial number of horses is found by setting n = 0 in the function: w ( 0 ) = 15 c d o t b 0 = 15 c d o t 1 = 15 .
The annual growth rate is calculated as b − 1 , where b = 1.35 , so the growth rate is 1.35 − 1 = 0.35 .
Convert the growth rate to a percentage by multiplying by 100: 0.35 c d o t 100 = 35% .
The initial number of horses is 15 and the annual percent growth rate is 35% , so the answers are 15 and 35 .
Explanation
Understanding the Problem We are given the function w ( n ) = 15 c d o t b n that models the population of horses on a ranch, where n is the number of years after Sabrina begins the research and b is an unknown base. We need to find the initial number of horses and the annual percent growth rate if b = 1.35 .
Finding the Initial Number of Horses To find the initial number of horses, we need to find the value of w ( n ) when n = 0 . So, we need to calculate w ( 0 ) = 15 c d o t b 0 . Since any number raised to the power of 0 is 1, b 0 = 1 . Therefore, w ( 0 ) = 15 c d o t 1 = 15 .
Calculating the Annual Percent Growth Rate To find the annual percent growth rate when b = 1.35 , we need to calculate the growth rate as a percentage. The growth factor is given by b = 1.35 . The growth rate is b − 1 = 1.35 − 1 = 0.35 . To express the growth rate as a percentage, we multiply by 100: 0.35 c d o t 100 = 35% .
Final Answer Based on the model, the initial number of horses is 15. If b = 1.35 , the annual percent growth rate of the number of horses would be 35% .
Examples
Understanding exponential growth, as modeled by Sabrina's horse population, is crucial in many real-world scenarios. For instance, it helps in predicting the spread of diseases, calculating investment returns, or even estimating the decay of radioactive materials. Just like the horse population grows over time, a viral marketing campaign might see an exponential increase in shares, or a savings account could grow exponentially with compound interest. Recognizing and analyzing exponential growth patterns allows us to make informed decisions and predictions in various aspects of life.
The initial number of horses is 15, and with a growth base of 1.35, the annual percent growth rate is 35%.
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