Answer :
Certainly! Let’s analyze the given exponential function and determine whether it represents growth or decay, along with the percentage rate of increase or decrease.
The given function is:
[tex]\[ y = 3700(0.97)^x \][/tex]
### Step-by-Step Solution:
1. Identify the Base of the Exponential Function:
The exponential term in the function is [tex]\(0.97\)[/tex].
2. Determine Growth or Decay:
- If the base (0.97 in this case) is less than 1, the function represents a decay.
- If the base is greater than 1, the function represents growth.
Since [tex]\(0.97\)[/tex] is less than 1, this means the function represents decay.
3. Calculate the Percentage Rate of Decrease:
- The rate of decay can be found by subtracting the base from 1 and then converting it to a percentage.
- Formula: [tex]\((1 - \text{base}) \times 100\)[/tex]
Plugging in the value:
[tex]\[
\text{Percentage rate of decrease} = (1 - 0.97) \times 100
\][/tex]
Simplifying this:
[tex]\[
= 0.03 \times 100
= 3 \%
\][/tex]
Therefore, the function represents a decay with a 3% rate of decrease.
The given function is:
[tex]\[ y = 3700(0.97)^x \][/tex]
### Step-by-Step Solution:
1. Identify the Base of the Exponential Function:
The exponential term in the function is [tex]\(0.97\)[/tex].
2. Determine Growth or Decay:
- If the base (0.97 in this case) is less than 1, the function represents a decay.
- If the base is greater than 1, the function represents growth.
Since [tex]\(0.97\)[/tex] is less than 1, this means the function represents decay.
3. Calculate the Percentage Rate of Decrease:
- The rate of decay can be found by subtracting the base from 1 and then converting it to a percentage.
- Formula: [tex]\((1 - \text{base}) \times 100\)[/tex]
Plugging in the value:
[tex]\[
\text{Percentage rate of decrease} = (1 - 0.97) \times 100
\][/tex]
Simplifying this:
[tex]\[
= 0.03 \times 100
= 3 \%
\][/tex]
Therefore, the function represents a decay with a 3% rate of decrease.
