Part (a):

We are given that 70% of the ocean region is ice-covered initially and the ice cover reduces by 6% per year.

After the first year, the percentage of the ice remaining will be: $\text{Percentage remaining} = 70\% \times (1 - 0.06)$ $\text{Percentage remaining} = 70\% \times 0.94 = 65.8\%$ Thus, after the first year, 65.8% of the region will be ice-covered.

Part (b):

To find the percentage of the region covered by ice after the third year, we need to reduce the ice cover by 6% for each year, compounded annually.

After the first year: $\text{Year 1} = 70\% \times 0.94 = 65.8\%$

After the second year: $\text{Year 2} = 65.8\% \times 0.94 = 61.852\%$

After the third year: $\text{Year 3} = 61.852\% \times 0.94 = 58.14088\%$ Rounding to the nearest tenth, the percentage of the region ice-covered after the third year is 58.1%.

Part (c):

We need to find how many years will pass before less than half (50%) of the region is covered by ice. This can be modeled by the equation: $P(t) = 70\% \times (0.94)^t$ where $P(t)$ is the percentage of the region covered by ice after $t$ years. We want to find $t$ when $P(t) < 50\%$.

Solving: $70 \times (0.94)^t = 50$ $(0.94)^t = \frac{50}{70} \approx 0.7143$ Now, take the natural logarithm of both sides: $\ln((0.94)^t) = \ln(0.7143)$ $t \cdot \ln(0.94) = \ln(0.7143)$ $t = \frac{\ln(0.7143)}{\ln(0.94)}$

Let me calculate that value for you.

It will take approximately 5.44 years for less than half of the region to be ice-covered. Since the number of years must be a whole number, this implies that after 6 years, the ice cover will drop below 50%.

Summary:

• (a) After the first year, 65.8% of the region will be ice-covered.
• (b) After the third year, 58.1% of the region will be ice-covered.
• (c) It will take approximately 6 years for less than 50% of the region to be ice-covered.

Would you like more details on any part of the solution?

Here are 5 related questions for practice:

1. What would the ice cover be after 10 years if the same 6% reduction continues annually?
2. If the annual reduction were 8%, how many years would it take for the ice cover to drop below 50%?
3. How much ice cover would remain after 5 years with a 5% annual reduction?
4. What is the compound percentage reduction over 4 years at 6% per year?
5. How would the result change if the initial ice cover were 80% instead of 70%?

Tip: When dealing with percentages and compounded rates, think of each year as multiplying by a reduction factor, which can be more accurate than subtracting the percentage directly each year.