We are given that initially 70% of the ocean region is ice-covered, and each year, the ice-covered region decreases by 6%. This type of situation represents exponential decay, where the percentage of ice left after each year can be modeled using the formula:

$A_n = A_0 \times (1 - r)^n$

where:

• $A_n$ is the percentage of the region covered in ice after $n$ years.
• $A_0$ is the initial percentage (70% in this case).
• $r$ is the annual reduction rate (6% or 0.06).
• $n$ is the number of years.

Part (a): Percent ice-covered after the first year

We need to calculate how much ice is left after the first year.

= 70 \times 0.94$$ $$A_1 = 65.8\%$$ So, after the first year, **65.8%** of the region will be ice-covered. --- ### Part (b): Percent ice-covered after the third year Now, we calculate the percentage of ice after three years. $$A_3 = 70 \times (1 - 0.06)^3 = 70 \times 0.94^3$$ Let me compute this. $$A_3 \approx 70 \times 0.830584$$ $$A_3 \approx 58.1\%$$ So, after the third year, **58.1%** of the region will be ice-covered. --- ### Part (c): Years before less than one-half of the region is covered We want to find how many years $$n$$ it will take for the ice to cover less than 50% of the region. We are solving for $$n$$ in the inequality: $$70 \times (1 - 0.06)^n < 50$$ Divide both sides by 70: $$(0.94)^n < \frac{50}{70}$$ $$(0.94)^n < 0.7143$$ Now, take the natural logarithm of both sides to solve for $$n$$: $$\ln(0.94^n) < \ln(0.7143)$$ $$n \ln(0.94) < \ln(0.7143)$$ Now solve for $$n$$: $$n > \frac{\ln(0.7143)}{\ln(0.94)}$$ Let me compute this. $$n \approx \frac{-0.3365}{-0.0619}$$ $$n \approx 5.4$$ Since $$n$$ must be a whole number, it will take about **6 years** before less than half of the region is covered by ice. --- ### Summary of results: a) After the first year: **65.8%** b) After the third year: **58.1%** c) Less than half the region will be covered in about **6 years**. Do you want more details or have any questions? Here are five related questions to deepen your understanding: 1. How would the ice-covered percentage change if the reduction rate were 8% per year? 2. What would be the percentage of ice left after 5 years? 3. How does changing the initial ice coverage affect the time it takes for less than half to be covered? 4. How can you model this situation using a different decay function, such as a piecewise linear function? 5. What environmental factors could cause the rate of ice reduction to vary from year to year? **Tip**: When dealing with exponential decay, always identify the initial value, the decay rate, and use logarithms to solve for time when needed!