. 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 by ice?

The problem involves exponential decay since the ice cover is reducing by a fixed percentage (6%) every year. Let's work through the three parts one by one:

### Given:
- Initial ice coverage: \(70\%\)
- Annual reduction: \(6\%\)
- This means that each year, \(94\%\) of the previous year's ice coverage remains.

#### General formula:
We can model the percentage of ice coverage after \(n\) years as follows:
\[
P_n = P_0 \times (0.94)^n
\]
Where:
- \(P_n\) is the percentage of ice coverage after \(n\) years.
- \(P_0\) is the initial percentage (70%).
- \(0.94\) is the fraction of ice remaining after each year (since it's reducing by 6% per year).

### Part (a) - Percent of region ice-covered after the first year:
For the first year (\(n = 1\)):
\[
P_1 = 70 \times (0.94)^1 = 70 \times 0.94 = 65.8\%
\]
So, after the first year, approximately **65.8%** of the region will be ice-covered.

### Part (b) - Percent of region ice-covered after the third year:
For the third year (\(n = 3\)):
\[
P_3 = 70 \times (0.94)^3
\]
Let's calculate this.

\[
P_3 = 70 \times (0.94)^3 = 70 \times 0.830584 = 58.1\%
\]
So, after the third year, approximately **58.1%** of the region will be ice-covered.

### Part (c) - When will less than half the region be covered by ice?

We need to find \(n\) such that \(P_n < 50\%\). This gives the equation:
\[
70 \times (0.94)^n < 50
\]
Dividing both sides by 70:
\[
(0.94)^n < \frac{50}{70} = 0.7143
\]
Now, take the natural logarithm (logarithms help solve equations involving exponents):
\[
\ln((0.94)^n) < \ln(0.7143)
\]
This simplifies to:
\[
n \times \ln(0.94) < \ln(0.7143)
\]
\[
n < \frac{\ln(0.7143)}{\ln(0.94)}
\]
Let's calculate the value of \(n\).

\[
n \approx \frac{\ln(0.7143)}{\ln(0.94)} \approx \frac{-0.3365}{-0.0619} \approx 5.44
\]
So, it will take approximately **6 years** for less than half of the region to be covered by ice.

---

### Summary of answers:
a) After the first year, approximately **65.8%** of the region will be ice-covered.
b) After the third year, approximately **58.1%** of the region will be ice-covered.
c) It will take approximately **6 years** for less than half the region to be covered by ice.

Would you like further details or explanations on any part of the solution? Here are some related questions:

1. How does exponential decay differ from linear decay?
2. What would the ice coverage be after 10 years?
3. How would the results change if the ice reduction rate were 4% instead of 6%?
4. What if the initial ice coverage was 80% instead of 70%? How would the results differ?
5. How can logarithms help solve other exponential decay problems?

**Tip**: Exponential decay problems often involve logarithms for solving equations when an unknown is in the exponent!

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 by ice?

Explore how an Arctic ice study predicts the reduction of ice coverage over time using exponential decay. Learn how to calculate the percentage of ice left after the first year, third year, and when it will drop below 50%. This problem involves exponential decay and logarithmic calculations, suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation