Question
create an exponential function to represent a population of 50 insects that is growing at a rate of 7%. which of the following functions is accurate?
a f(x)=50(0.070^x
b f(x)=50(1.07)^x
c f(x)=50(1.7)^x
d f(x)=50(0.93)^x
a f(x)=50(0.070^x
b f(x)=50(1.07)^x
c f(x)=50(1.7)^x
d f(x)=50(0.93)^x
Answers
GPT-4o mini
To create an exponential function that represents a population of 50 insects growing at a rate of 7%, we can use the standard formula for exponential growth:
\[ f(x) = P_0 \cdot (1 + r)^x \]
where:
- \( P_0 \) is the initial population,
- \( r \) is the growth rate (expressed as a decimal),
- \( x \) is the time in appropriate time units,
- \( 1 + r \) represents the growth factor.
In this case:
- \( P_0 = 50 \)
- \( r = 0.07 \)
Putting this into the formula, we have:
\[ f(x) = 50 \cdot (1 + 0.07)^x \]
\[ f(x) = 50 \cdot (1.07)^x \]
Now, we can look at the options provided:
- a) \( f(x) = 50(0.070^x) \)
- b) \( f(x) = 50(1.07^x) \)
- c) \( f(x) = 50(1.7^x) \)
- d) \( f(x) = 50(0.93^x) \)
The correct exponential function representing the population growing at a rate of 7% is:
**b) \( f(x) = 50(1.07^x) \)**
\[ f(x) = P_0 \cdot (1 + r)^x \]
where:
- \( P_0 \) is the initial population,
- \( r \) is the growth rate (expressed as a decimal),
- \( x \) is the time in appropriate time units,
- \( 1 + r \) represents the growth factor.
In this case:
- \( P_0 = 50 \)
- \( r = 0.07 \)
Putting this into the formula, we have:
\[ f(x) = 50 \cdot (1 + 0.07)^x \]
\[ f(x) = 50 \cdot (1.07)^x \]
Now, we can look at the options provided:
- a) \( f(x) = 50(0.070^x) \)
- b) \( f(x) = 50(1.07^x) \)
- c) \( f(x) = 50(1.7^x) \)
- d) \( f(x) = 50(0.93^x) \)
The correct exponential function representing the population growing at a rate of 7% is:
**b) \( f(x) = 50(1.07^x) \)**