Question

Scientists have found the population of insects in a certain area has increased by
12.5% each year since 2020. The initial insect population recorded in 2020 was
450 insects.

Drag the correct value to each box to complete the function that represents this situation.
Drag the correct value to the box to identify, to the nearest whole number, the number of insects there will be in this population in the year 2030.

f(x)=___(___)^x
f(10)=___

Answer 1

To represent the population growth of insects in a mathematical function, we can use the general formula for exponential growth:

\[
f(x) = P_0 (1 + r)^x
\]

where:
- \( P_0 \) is the initial population,
- \( r \) is the growth rate (as a decimal),
- \( x \) is the number of years since the initial population was recorded.

In this scenario:
- The initial population in 2020, \( P_0 = 450 \).
- The annual growth rate is \( 12.5\% \), which in decimal form is \( r = 0.125 \).

Now we can plug these values into the function:

\[
f(x) = 450(1 + 0.125)^x
\]
\[
f(x) = 450(1.125)^x
\]

Now, we need to find the population in the year 2030. Since 2020 is year 0, 2030 will be year 10. So, we need to evaluate \( f(10) \):

\[
f(10) = 450(1.125)^{10}
\]

Now, calculating \( (1.125)^{10} \):

\[
(1.125)^{10} \approx 3.207135472
\]

Then we multiply this by 450:

\[
f(10) \approx 450 \times 3.207135472 \approx 1443.21
\]

Rounding this to the nearest whole number gives us approximately 1443.

To complete fill-in:

1. The full function is:
\[
f(x) = 450(1.125)^x
\]

2. The estimated population in 2030 is approximately:
\[
f(10) = 1443
\]

Thus:

- First box: **450 (1.125)**
- Second box: **1443**