Problem 44 - Algebra 2

Question

ART A one-ton ice sculpture is melting so that it loses one-eighth of its weight per hour. How much of the sculpture will be left after five hours? bWrite your answer in pounds.

Step-by-Step Solution

Answer

750 pounds

Step 1: Understand the problem

The ice sculpture originally weighs one ton, equivalent to 2000 pounds. It loses one-eighth of its weight per hour. The task is to find out how much of the sculpture will be left after five hours.

Step 2: Calculate hourly weight loss

Since the sculpture loses one-eighth of its weight every hour, in the first hour, it will lose \(\frac{1}{8} \times 2000 = 250\) pounds.

Step 3: Compute the remaining weight after each hour

Starting with 2000 pounds, subtract 250 pounds each hour: - After 1st hour: 2000 - 250 = 1750 pounds - After 2nd hour: 1750 - 250 = 1500 pounds - After 3rd hour: 1500 - 250 = 1250 pounds - After 4th hour: 1250 - 250 = 1000 pounds - After 5th hour: 1000 - 250 = 750 pounds.

Step 4: Conclude the remaining weight

After five hours, the remaining weight of the ice sculpture is 750 pounds.

Key Concepts

Understanding the Rate of Change

The rate of change is a fundamental concept in mathematics that helps us understand how quickly or slowly a certain quantity is changing over time. In this exercise, the one-ton ice sculpture loses one-eighth of its weight every hour.

This means that the **rate of change** is constant and negative, as the sculpture is decreasing in weight.

• Initially, the sculpture weighs 2000 pounds (since one ton equals 2000 pounds).
• Every hour, it loses a part of its weight, specifically one-eighth.

By calculating this change consistently over a period, we can predict how much of the sculpture remains. The formula to find the hourly weight loss is straightforward:

Let the initial weight be denoted as \( W_0 = 2000 \text{ pounds} \), and the rate of weight loss each hour as \( \frac{1}{8} \).

• Hourly weight loss: \( \frac{1}{8} \times W_0 \)
• For our sculpture, it equates to \( \frac{1}{8} \times 2000 = 250 \text{ pounds} \) per hour.

Understanding and applying the rate of change is essential for calculations like these in real life.

Calculating Weight Loss

Weight loss calculation does not just apply to dieting; it also applies here in determining how much of an object remains after a certain time.

In this ice sculpture scenario, we need to keep removing a fixed amount from the total for every hour that passes.

• After the first hour: We subtract 250 pounds (the hourly loss) from 2000 pounds to get 1750 pounds left.
• After the second hour: Subtract another 250 pounds, resulting in 1500 pounds.
• This process continues hourly, ultimately leaving 750 pounds after five hours.

By calculating the weight loss step by step, we ensure accuracy in our final result.

This methodical reduction helps avoid any errors that could occur if different rates were applied. It's crucial to follow the process precisely, especially if multiple factors could influence the object beyond its basic rate of change.

Effective Step-by-Step Problem Solving

Step-by-step problem solving is essential for breaking down complex problems into manageable, easy-to-follow stages.

In the context of the melting sculpture, here's how such an approach benefits us:

• **Understanding the Problem Completely:** Knowing the initial weight and rate of change gives a clear starting point.
• **Performing Calculations Systematically:** By subtracting the same amount each hour, we maintain consistency, which makes verification straightforward.
• **Documenting Each Step:** This improves comprehension and ensures the process can be reviewed if needed._

Such an approach not only helps in solving mathematical problems but also develops logical thinking and precision in real-life scenarios.
By breaking down a task into smaller parts, it becomes easier to manage and solve efficiently, leading to a correct and reliable outcome.