Question

the number of bee colonies worldwide was estimated to be 101.6 million in 2021. this number is decreasing by about 40% each year due to disease, parasites, pesticides, and more factors that are causing the number of bees to die worldwide. bees are vital to our environment because they pollinate our flowers and trees, produce honey and beeswax used in materials found in household products and medications, and are an important component in the animal/insect food chain.

1. write an exponential equation to model the number of bee colonies y, in millions, based on the number of years since 2021, x.
2. use your model to estimate the number of bee colonies for the year 2024.
3. if this trend continues, how many years would it take for the number of bee colonies to reach only 5 million?
4. what can be done to help the bee population start to increase again?

Explanation:

Step1: Define exponential decay formula

The general form of an exponential decay model is $y = a(1-r)^x$, where $a$ is the initial amount, $r$ is the decay rate, and $x$ is time.

Step2: Plug in known values for Q1

Initial amount $a=101.6$, decay rate $r=0.40$, so $1-r=0.60$.
Expression: $y = 101.6(0.60)^x$

Step3: Calculate x for 2024 (Q2)

2024 is $2024-2021=3$ years after 2021, so $x=3$.
Expression: $y = 101.6(0.60)^3$

Step4: Compute the value for Q2

First calculate $(0.60)^3=0.216$, then multiply by 101.6:
Expression: $y = 101.6 \times 0.216 = 21.9456$

Step5: Set up equation for Q3

Set $y=5$, solve for $x$: $5 = 101.6(0.60)^x$

Step6: Isolate the exponential term (Q3)

Divide both sides by 101.6:
Expression: $\frac{5}{101.6} = (0.60)^x$
$\frac{5}{101.6} \approx 0.0492$

Step7: Solve for x using logarithms (Q3)

Take natural log of both sides: $\ln(0.0492) = x\ln(0.60)$
Solve for $x$:
Expression: $x = \frac{\ln(0.0492)}{\ln(0.60)} \approx \frac{-3.013}{-0.5108} \approx 5.9$

Step8: List solutions for Q4

Identify evidence-based methods to support bee populations.

Answer:

1. $y = 101.6(0.60)^x$
2. 21.95 million (rounded to two decimal places)
3. Approximately 6 years
4. - Reduce or eliminate the use of harmful pesticides

• Plant native, bee-friendly flowers and vegetation to provide food sources
• Create bee nesting habitats (like bee hotels)
• Support local beekeepers through purchasing local honey
• Implement policies to protect pollinator habitats from urban development