. 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

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.

\[
A_1 = 70 \times (1 - 0.06)^1
= 70 \times 0.94
\]

\[
A_1 = 65.8\%
\]

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!

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?

Solve a problem on Arctic ice coverage with a 6% annual reduction rate. Calculate the percentage of ice after 1 year, 3 years, and determine how long it will take for less than half of the region to be ice-covered. This problem explores exponential decay and logarithmic equations, suitable for Grades 9-12.

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