Question

Rodrigo is an ichthyologist, or a scientist who studies fish. When he first started monitoring the sturgeon population of Lavender Lake, he estimated there were 600 sturgeon in the lake. One year later, Rodrigo estimated the population had decreased to 510. He expects the sturgeon population to continue decreasing each year. Write an exponential equation in the form y=a(b)x that can model the estimated sturgeon population, y, in x years. Use whole numbers, decimals, or simplified fractions for the values of a and b.

Answer 1

To write an exponential equation that models the estimated sturgeon population in Lavender Lake, we can start by identifying the parameters needed for the equation in the form \( y = a(b)^x \).

1. **Identify the initial population, \( a \)**:
- At year 0 (when \( x = 0 \)), the population \( y \) is 600 sturgeon. Therefore, \( a = 600 \).

2. **Identify the population after one year**:
- After one year (when \( x = 1 \)), the population \( y \) is 510 sturgeon.

3. **Set up the equation with the information we have**:
- We know that after 1 year, \( y \) can be expressed as follows:
\[
y = 600(b)^1
\]
- And since \( y = 510 \) when \( x = 1 \):
\[
510 = 600b
\]

4. **Solve for \( b \)**:
\[
b = \frac{510}{600}
\]
\[
b = \frac{51}{60} = \frac{17}{20}
\]

5. **Write the final equation**:
- Now, we can put \( a \) and \( b \) together in our exponential equation:
\[
y = 600\left(\frac{17}{20}\right)^x
\]

Therefore, the exponential equation that models the estimated sturgeon population \( y \) in \( x \) years is:
\[
y = 600\left(\frac{17}{20}\right)^x
\]