Problem 86 - Algebra A Combined Function

Question

A glacier is a giant mass of rocks and ice that flows downhill like a river. Exit Glacier, near Seward, Alaska, moves at a rate of 20 inches a day. Find the distance in feet the glacier moves in a year. (Assume 365 days a year.) Round to two decimal places.

Step-by-Step Solution

Answer

The glacier moves approximately 609.55 feet per year.

Step 1: Convert Inches to Feet

Since 1 foot is equal to 12 inches, we need to convert the glacier's daily movement from inches to feet. To do this, divide 20 inches by 12. So, the glacier moves \( \frac{20}{12} \) feet per day.

Step 2: Calculate Daily Movement in Feet

Divide the number of inches the glacier moves per day by 12 to find the movement in feet. This is: \( \frac{20}{12} \approx 1.67 \) feet per day.

Step 3: Calculate Yearly Movement in Feet

Multiply the glacier's daily movement in feet by the number of days in a year (365). This gives: \( 1.67 \times 365 = 609.55 \) feet per year.

Key Concepts

Glacier Movement

Glaciers are incredible natural phenomena that slowly glide over land. They are made up of compacted snow and ice, often gathering rocks and debris as they move. Unlike fast-moving rivers, glaciers shift much more slowly, sometimes only inches or feet in a day.
This slow, steady movement is determined by the force of gravity, which pulls on the ice mass, causing it to flow downhill. This is similar to how thick honey might gradually move down a surface. Understanding glacier movement helps us to analyze various environmental factors, such as climate change and glacial erosion.
In the context of our exercise, knowing the rate of movement (20 inches per day) is crucial for calculations related to distance over time.

Inches to Feet Conversion

In everyday life, we often need to convert measurements from one unit to another. This is important in science, engineering, and even cooking. The key is to know the conversion factor—in this case, the relationship between inches and feet.
One foot is equal to 12 inches. Therefore, to convert inches to feet, you simply divide the number of inches by 12.

• For example, if you have 24 inches, dividing by 12 will give you 2 feet.
• In our exercise, the glacier moves 20 inches a day. So, you convert this by calculating: \( \frac{20}{12} \approx 1.67 \) feet per day. It's as simple as that!

Understanding this concept is very useful for those engaged in tasks that require precise measurement and comparison of lengths.

Distance Calculation

Once you know how far something moves in a day, calculating the total movement over a longer period, like a year, becomes straightforward. This is often referred to as a distance calculation.
To do this in our exercise, we take the daily movement in feet and multiply it by the number of days in the time period (365 days for a year).

• This transforms our daily estimate to a yearly figure: \( 1.67 \times 365 = 609.55 \) feet.

Being able to perform such calculations allows you to understand real-world scenarios better—whether you’re tracking the movement of glaciers or estimating travel distances.

Algebraic Calculations

Algebra helps in structuring and solving equations that predict real-life occurrences. The arithmetic of converting daily movements to yearly totals is rooted in basic algebraic calculations and understanding.
In our example, we used a simple equation. We first needed to know the daily foot movement ((\frac{20}{12}]}, and then multiply that by 365. The equivalent algebraic expression is:\[ y = x \times 365 \]
where \(x\) is daily movement in feet and \(y\) is the total annual movement in feet.

• Such calculations make it straightforward to handle more complex problems as well, by using a series of manageable steps.
• They form the basis of more advanced mathematics and practical problem solving.

Learning these methods strengthens your ability to tackle mathematical challenges in various domains with confidence.