. An Arctic ice study found that in the
summer, 70% of an ocean region was ice
covered. The study also predicted the
region’s ice would be reduced by 6%
per year.
a) What percent of the region will be ice
covered after the fi rst year? Express
your answer to the nearest tenth of
a percent.
b) What percent of the region will be ice
covered after the third year? Express
your answer to the nearest tenth of
a percent.
c) How many years will pass before less
than one half of the region is covered

### 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.

An Arctic ice study found that in the summer, 70% of an ocean region was ice-covered. The study also predicted the region’s ice would be reduced by 6% per year. a) What percent of the region will be ice-covered after the first year? Express your answer to the nearest tenth of a percent. b) What percent of the region will be ice-covered after the third year? Express your answer to the nearest tenth of a percent. c) How many years will pass before less than one half of the region is covered?

Explore how Arctic ice coverage decreases over time with a 6% annual reduction rate. Learn how to calculate the percentage of ice cover after the first and third years, and find out how long it will take for the ice to drop below 50%. Suitable for students in Grades 9-11.

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